Embed Size (px)
DESCRIPTION
Sequencias e séries numéricas
Citation preview
!"#$%&' (
)*+,-. /0) * )12/*)
!"#$%&'( )*' +,*- .' /*01$2-' #(0#3*4(# 52# ' *-2,' (#"* /*0*6 .#78
9: ;#/',<#/#3 2=* (#52>,/%* # %+/*38
)*7 (# ? /',@#,$# '2 .%@#,$#A
)!7 (# ? /3#(/#,$# '2 .#/3#(/#,$#A
)/7 03'03%#.*.#( .# 2=* (#52>,/%*:
B: C#+,%3 (?3%#( ,2=?3%/*( .# $#3='( 0'(%$%&'(A
D: E,/',$3*3 * ('=* .# (?3%#(A
F: G.#,$%+/*3 *( (?3%#( #(0#/%*%(8 @#'=?$3%/*H <*3=I,%/*H (?3%#40A
J: K#3%+/*3 (# * (?3%# ? /',@#,$# '2 .%@#,$#H *0-%/*,.' '( /3%$?3%'( .# /',@>,/%*A
L: M,*-%(*3 * /',@>,/%* .# (?3%#( *-$#3,*.*( # .# (%,*%( 52*%(52#3A
N: ;#/',<#/#3 (?3%#( *!('-2$*=#,$# # /',.%/%',*-=#,$# /',@#,$#(A
O: ;#/',<#/#3 (?3%#( .# P2,QR#(A
S: E,/',$3*3 ' 3*%' # ' %,$#3&*-' .# /',@>,/%* .*( (?3%#( .# 0'$>,/%*(A
9T: C#(#,&'- P2,QR#( #= (?3%#( .# U*V-'3 # W*/-*23%,A
99: X$%-%6*3 (?3%#( .# P2,QR#( ,* 3#('-2QY' .# -%=%$#( # %,$#@3*%(A
9B: ;#('- #Z#3/1/%'( 2(*,.' 2=* P#33*=#,$* $#/,'-[@%/*:
M 03'&* (#3\ /'=0'($* 0'3 52#($R#( 52# 0'((%!%-%$*= %+/*3 (# '( '!"#$%&'( P'3*=
*$%,@%.'(: ]'3$*,$'H #((# ? ' 3'$#%3' 0*3* '3%#,$*QR#( .# (#2( #($2.'(: ='.#-' .# P'3=24
-*QY' .*( 52#($R#( ? ' ='.#-' *.'$*.' ,* P'3=2-*QY' .'( #Z#3/1/%'( # ,' .#(#,&'-&%=#,$'
$#[3%/' .#((# /*01$2-'H ,#((* *0'($%-*:
9LS
!" #$%&'()*+'
!"#! $%&'#()* !"#(+%,!-*" ".,/!" /010/#%"2 %" 3(%/" "4* "*-%" 3(! !05*)5!- (- 06-!,*
/010/#* +! #!,-*"7 8" ".,/!" /010/#%" +!"!-&!09%- (- &%&!) :(0+%-!0#%) #%0#* 0% -%#!-;#/$%
3(%0#* 0% $/<0$/%7 =)%" "4* ("%+%"2 &*, !>!-&)*2 &%,% %&,*>/-%, :(0?@!" #,/A*0*-.#,/$%"
! )*A%,'#-/$%"2 &%,% ,!"*)5!, !3(%?@!" +/:!,!0$/%/"2 &%,% !:!#(%, /0#!A,%/" $*-&)/$%+%"2 &%,%
$,/%, 0*5%" :(0?@!" ! &%,% $*0"#,(/, -*+!)*" -%#!-;#/$*" +! )!/" :'"/$%" B80#*02 CDDDE7
!, -./)0$1234
% )/0A(%A!- $*#/+/%0%2 * #!,-* "!3(<0$/% "/A0/1$% (-% "($!""4* +! $*/"%" !- (-% *,+!-
+!#!,-/0%+% *,+!- $,*0*)FA/$%2 +! #%-%09*2 *( )FA/$%2 &*, !>!-&)*7 =- -%#!-;#/$% *
#!,-* "!3(<0$/% . ("%+* $*-(-!0#! &%,% +!0*#%, (-% "($!""4* +! 06-!,*" $(G% *,+!- .
+!#!,-/0%+% &*, (-% )!/ *( :(0?4*7
="#(+%,!-*" (- #/&* !"&!$/%) +! :(0?4* +!10/+% 0*" 06-!,*" 0%#(,%/" N∗ = {1, 2, 3, 4, · · · }$*- /-%A!- !- R. H"#* .2 !"#(+%,!-*" % :(0?4* f : N∗ → R 3(%0#* %* )/-/#! ! "(%" &,*I
&,/!+%+!" 3(%0+* n→∞. 8 :(0?4* f : N∗ → R +!10/+% &*, f(n) = n2n+1
. (- !>!-&)* +!
"!3(<0$/%7 J $*0G(0#* $*-&*"#* &!)*" &%,!" *,+!0%+*" (n, f(n))2 +%+* &*,
I = {(1, f(1)), (2, f(2)), (3, f(3)), · · · , (n, f(n)), · · · }
*(
I =
{
(1,1
3), (2,
2
5), (3,
3
7), · · · , (n, n
2n+ 1), · · ·
}
. +!0*-/0%+* $*0G(0#* +*" #!,-*" +% "!3(<0$/% f(n). K!,%)-!0#!2 * $*0G(0#* I . !"$,/#*
+! :*,-% "/-&)/1$%+%7 H"#* .2 I . ,!&,!"!0#%+* &!)%" /-%A!0" +! n ∈ N∗ +! :*,-% 3(! %&*"/?4* 3(! +!#!,-/0%+% /-%A!- +! f *$(&% 0* $*0G(0#* +*" #!,-*" +% "!3(<0$/% f(n) .+!#!,-/0%+% &!)* !)!-!0#* n ∈ N∗, *( "!G%2
I = {f(1), f(2), f(3), · · · , f(n), · · · } ={
1
3,2
5,3
7,4
9,5
11, · · · , n
2n+ 1, · · ·
}
.
L*+!-*" *M"!,5%, 3(! * #!,-*
511. /-%A!- +! n = 5, &*/" *$(&% % 3(/0#% &*"/?4* 0*
$*0G(0#* +*" #!,-*"7 J #!,-* f(n) = n2n+1
. +!0*-/0%+* #!,-* A!,%) +% "!3(<0$/%7 8
:*,-% ("(%) +! ,!&,!"!0#%, * #!,-* A!,%) +! (-% "!3(<0$/% . un = n2n+1
*( xn = n2n+1
*(
yn = n2n+1
!#$7 L%""%,!-*" %A*,% N +!10/?4* :*,-%) +! "!3(<0$/%7 !""! $%"*2 #!-*" *
$*0G(0#* I = {u1, u2, u3, · · · , un, · · · }.
!"#"$%& '()(* !"#$ N∗ = {1, 2, 3, 4, · · · } % &%'"(')% *%+ '#)(,#-+. R # ,!)# ,!#/0 1!2
'%$-'#$%+ # #3/--% un : N∗ → R *! ($# +!6(7'&-# '($8,-�
!"#$% &'('( 9#,# $!/:%, &%$3,!!'+5%. ;#$%+ +(3%, 6(! % &,!+&-$!')% *-<,-% *! ($#
/-':#=!$ *! +(>'%+ 8 *#*# !$ ?('45% *% &,!+&-$!')% )%)#/ 3!/# +!6(7'&-# un = nn+13
%'*!
n &%,,!+3%'*! #% '@$!,% *! *-#+ *! ;-*# *% +(>'% ! limn→∞
un % )#$#':% *! ($ +(>'% #*(/)%0
A++-$. % &%'"(')%
{
114, 215, 316, 417, 518, · · · , n
n+13, · · ·
}
,!3,!+!')# % )#$#':% *-<,-% *% +(>'% !$
,!/#45% #% )#$#':% B'#/0
K,%1$%-!0#! &*+!-*" *M"!,5%, % $(,5% +! $,!"$/-!0#*2 $(G* )/-/#! . ,!&,!"!0#%+* &!)%
%""'0#*#% y = 1 BO/A(,% P7CE7
CQR
!"#$% &'() *$+,-!.+/01 2% 3!/4%"+. 2+ ,#5/1,
*1.1 612+.1, 17,+$8%$ % %,,5/010% y = 1 $+6$+,+/0% 1 3!.!0+ 2+ -$+,-!.+/01 21 ,#5/1'9,,1 ,!"/!:-% ;#+ 612+.1, 3+8%/0%$ ;#+,0<+, -1.1 61$ +=+.631> ;#%3 1 /?.+$1 .5/!.1 2+
2!%, ;#+ 1 ,#5/1 2+8+ :-%$ +. 0$%0%.+/01 6%$% %0!/"!$> 6+31 .+/1,> 80% 2+ ,+# 0%.%/41
:/%3@
A1 !"#$% &'B 612+.1, 17,+$8%$ #.% +,0!.%0!8% +. 01$/1 2+ &C 2!%,'
!"#$% &'B) D,0!.%0!8% 6%$% 170+$ EC 61$ -+/01 21 0%.%/41 :/%3
F ;#+,0G1 %"1$% H) -1.1 I%J+$ #.% +,0!.%0!8% +. 0+$.1, .%0+.K0!-1,@ F $+,61,0% ,+$K
2%2% 6+3% 2+:/!LG1 2+ 3!.!0+ 2+ #.% ,+;#M/-!%'
!"!# $%&%'( )( *&+ ,(-*./0%+
!"#"$%& '()(* !"# un $%# &!'$()*+#, -+.!%/& '$! / )0%!1/ a 2 3+%+4! -! un '$#)-/
n 4!)-! 5#1# / +)6)+4/ &!, -#-/ ε > 0 5/-!%/& !)*/)41#1 K > 0 4#3 '$! 5#1# 4/-/ n > K
7#3! # -!&+8$#3-#-! |un − a| < ε.
!"#$% &'('& 9#-# # &!'$()*+# un : N∗ → R -!6)+-# )/ :;!%53/ <=>=> 5/1 un = nn+13
,
7#%/& %/&41#1 '$! lim un = 1.
!"#$%&"' N+8+.1, .1,0$%$ ;#+> 2%21 ε > 0 612+.1, +/-1/0$%$ K > 0 0%3 ;#+ 6%$% 0121n > K 8%3+ % 2+,!"#%32%2+ |un − a| < ε. F"1$%>
|un − 1| =∣
∣
∣
∣
n
n+ 13− 1
∣
∣
∣
∣
=
∣
∣
∣
∣
n− n− 13
n+ 13
∣
∣
∣
∣
=
∣
∣
∣
∣
13
n+ 13
∣
∣
∣
∣
< ε.
(O(
! "#$# %&! '#$!"#( !()*!+!*
13
n+ 13< ε ⇒ 13 < nε+ 13ε ⇒ 13− 13ε
ε< n.
,#-(!%&!-.!"!-.!/ '#$!"#( .#"0* K = 13−13εε
! 0 !1-234# 56768 !(.0*9 (0.2(:!2.06
,#"'0*0-$# #( $0$#( $# ;<!"'=# 56767 )#" 0 !1-234# 56768 )#-)=&>"#( %&! ε = 0, 2*!'*!(!-.0 0 $2:!*!-30 !-.*! # )*!()2"!-.# 0="!?0$# ! # )*!()2"!-.# .#.0= $#( (&>-#(6 @#*
#&.*# =0$#/ K A # -B"!*# ">-2"# $! $20( %&! #( (&>-#( $!+!" '!*"0-!)!* !" .*0.0"!-.#
'0*0 0.2-C2*/ '!=# "!-#(/ 80% $! (!& )*!()2"!-.# .#.0=6
!"#$% &'(') !"!#$%&! ' &($!#' $)&%$' *! *%+, -.! .$ /'"! *! ,.)&',0 1.2' 1#!,1%$!&"'
3 *+*' 4!/+ ,!-.5&1%+ un = nn+13
*!6! 4!#$+&!1!# !$ "#+"+$!&"' 4+#+ +"%&7%#0 #!,4!1"%6+8
$!&"!0 80%, 90% ! 95% *' ,!. "+$+&9' :&+/;
!"#$%!& D# ;<!"'=# 56765 )#-)=&>"#( %&! $0$# ε > 0 '#$!"#( .#"0* K = 13−13εε
. ,#"#
'0*0 80%, 90% ! 95% $# .0"0-E# 1-0= #( +0=#*!( $! ε (4# *!('!).2+0"!-.! 0.2, 0.1 !
0.05 .!"#(/ *!('!).2+0"!-.!/ # -B"!*# ">-2"# $! $20( A $0$# '#*
(a) K =13− 13ε
ε=
13− 13 · 0, 20, 2
= 52 $20(
(b) K =13− 13ε
ε=
13− 13 · 0, 10, 1
= 117 $20(
(c) K =13− 13ε
ε=
13− 13 · 0, 050, 05
= 247 $20(
F&.*0 )#-)=&(4# %&! '#$!"#( .2*0* A %&!/ 0 '0*.2* $! &" $!.!*"2-0$# .!"'#/ 0 +0*2034#
$# )*!()2"!-.# A "&2.# '!%&!-0 !" *!=034# G %&0-.2$0$! $! *034# %&! # (&>-# )#-(#"!6
@#*.0-.#/ # '*#$&.#* $!+! !(.2"0* # .!"'# ">-2"# $! .*0.0"!-.# !" $20( '0*0 #H.!* #
"9<2"# $! =&)*#6
!"!# $%&'()*+,- ./)0%12%)3%-
' !"#"$%& '()(* <!2+ un .$+ ,!-.5&1%+; %=!$', -.! un 3 1'&6!#7!&"! ,!0 ! ,'$!&"! ,!0
limn→∞
un = L 4+#+ +/7.$ L ∈ R.
I! un -4# :#* )#-+!*C!-.!/ $2*!"#( %&! un ( )*+,-.,/0,6
!"#$% &'('* > ,!-.5&1%+ un = 2n+33n+5
3 1'&6!#7!&"!0 4'%, limn→∞
un = limn→∞
2n+33n+5
= 23.
!"#$% &'('+, !"!#$%&! ,! + ,!-.5&1%+ un = 14n2 − 1 1'&6!#7! '. *%6!#7!;
!"#$%!& J (!%&K-)20 $0$0 A .0= %&! limn→∞
un = limn→∞
14n2 − 1 =∞.
,#"# # =2"2.! $! un -4# !<2(.!/ 0 (!%&K-)20 $2+!*C!6
4 !" #$ %&'&(( <!2+ un : N∗ → R .$+ ,!-.5&1%+ !$ R "+/ -.! limn→∞
un !?%,"!0 !&"@'
!,"! /%$%"! 3 (&%1';
)!"%-./0123%4 I&'#-E0"#( %&! un : N∗ → R A &"0 (!%&K-)20 !" R .0= %&! limn→∞
un
!<2(.! ! (&'#-E0"#( %&! a ! b, )#" a 6= b, (4# =2"2.!( $!((0 (!%&K-)206 ;-.4# $0$# ε > 0'#$!"#( !-)#-.*0* K1 > 0 ! K2 > 0 .0= %&! '0*0 .#$# n > K1 .!-E0"#( |un − a| < ε
2!
'0*0 .#$# n > K2 .!-E0"#( |un − b| < ε2. JC#*0 (!?0 K = max{K1, K2}. ;-.4# '#$!"#(
!()*!+!*/ '0*0 .#$# n > K
LM7
|a− b| = |a− un + un − b| = |−(un − a)− (un − b)|≤ |un − a|+ |un − b| < ε
2+ ε
2= ε.
!"! a # b $%! &!'$()'(#$* (#+#"!$ |a− b| < ε ,)+) (!-! ε > 0 $#* # $!"#'(# $#|a− b| = 0, .$(! /* $# a = b. 0!1!* ! 2.".(# -# un, $# #3.$(#* / 4'.&!5
!" #$%&'($)*+,-&
!"#"$%& '()(* !"# un : N∗ → R $%# &!'$()*+#, !"# N ′ = {n1 < n2 < n3 < · · · <nk < · · · } $% &$-*.)"$)/. +)0)+/. 1! N∗, !)/2. unk
= un
∣
∣
N ′: N∗ → R 3 1+/# $%# &$-&!4
'$()*+# 1! un.
!"#$% &'(') !"# un : N∗ → R $%# &!'$()*+# 1#1# 5.6 un = 1n2 . !"# N
′ = {1, 3, 5, 7, · · · } ⊂N∗. 7)/2. # &!'$()*+# unk
: N ′ → R 3 $%# &$-&!'$()*+# 1! un. 8& /!6%.& 1# &!'$()*+# &2.
{1, 14, 19, 116, 125, 136, 149, · · · } ! .& /!6%.& 1# &$-&!'$()*+# &2. {1, 1
9, 125, 149, · · · }.
!" #$ %&'&' ! $%# &!'$()*+# *.)9!6:! 5#6# L, !)/2. /.1#& &$#& &$-&!'$()*+#& /#%4
-3% *.)9!6:!% 5#6# L.
(!"%*+,-./0%1 67,!'8)"!$ 97# un : N∗ → R / 7") $#97:'&.) ()2 97# limn→∞
un = L. ;$$."*
-)-! ε > 0, #3.$(# K > 0 ()2 97# ,)+) (!-! n > K / <=2.-) ) -#$.17)2-)-# |un − L| < ε.
;1!+)* $# unk: N ′ → R / 7") $7>$#97:'&.) -# un, !'-# N
′ = {n1 < n2 < · · · < nk < · · · }/ 7" &!'?7'(! .'@'.(!* (#"!$ 97#* ,)+) &)-) ε > 0, #3.$(# 7" k0 ∈ N∗ ()2 97# nk0 > K #
#'(%!* ,)+) k > k0 (#"!$ 97# nk > nk0 > K # )$$." |unk− L| < ε, ! 97# ,+!<) 97# unk
()">/" &!'<#+1# ,)+) L, &!"! 97#+A)"!$ -#"!'$(+)+5
!"#$% &'('2 ; &!'$()*+# un = (−1)n 3 1+9!6:!)/!< 5.+& #1%+/! &$-&!'$()*+#& '$! *.)4
9!6:!% 5#6# 9#=.6!& 1+>!6!)/!&< *.)/6#6+#)1. . /!.6!%# #)/!6+.6, ?! >#/.< # &$-&!'$()*+# 1!
@)1+*!& 5#6!&< 1#1# 5.6 u2n = (−1)2n = 1 *.)9!6:! 5#6# L1 = 1, !)'$#)/. '$! &$# &$-&!4
'$()*+# 1! @)1+*!& @%5#6!&< 1#1# 5.6 un = (−1)2n+1 = −1 *.)9!6:! 5#6# L2 = −1. A.%. .&
=+%+/!& 1#& &$-&!'$()*+#& &2. 1+>!6!)/!&< # &!'$()*+# 1+9!6:!,
!. #'($)*+,- /,0,1-2-
!"#"$%& '(+(* !"# un : N∗ → R $%# &!'$()*+# !% R. ?+B!%.& '$! un 3 =+%+/#1# &!
. *.)"$)/. {u1, u2, u3, · · · , un · · · } >.6 =+%+/#1.< .$ &!"#< &! !C+&/+6!% k1 ! k2 ∈ R /#+& '$!
k1 ≤ un ≤ k2 5#6# /.1. n ∈ N∗.
!" #$ %&)&* !"# un : N∗ → R $%# &!'$()*+# *.)9!6:!)/! !% R, !)/2. un 3 =+%+/#1#,
(!"%*+,-./0%1 67,!'8)"!$ 97# un : N∗ → R / 7") $#97:'&.) &!'<#+1#'(# #" R #
$7,!'8)"!$ 97# a / 2.".(# -#$$) $#97:'&.)5 B'(%!* -)-! ε = 1, ,!-#"!$ #'&!'(+)+ K > 0,()2 97# ,)+) (!-! n > K (#'8)"!$ |un − a| < 1. ;$$."* ,)+) (!-! n > K, (#"!$ un ∈B(a, 1). !"! ! &!'?7'(! {u1, u2, u3, · · · , uK} / @'.(!* 2!1! )-".(# 7" <)2!+ "=3."!* $#?)
M = max u1, u2, · · · , uK , $#17# 97# {u1, u2, u3, · · · , un−1, un, · · · } ⊂ B(a, 1)∪B(0,M). 0!1!*un / 2.".()-)5
!3+!-4./0% &'2'( ; 6!*@56.*# 1!&&! /!.6!%# )2. 3 9!61#1!+6#, D.6 !C!%5=.< un = (−1)n 3
=+%+/#1#< *.% −1 ≤ un ≤ 1, %#& un )2. 3 *.)9!6:!)/!,
CDE
! "#$%&'()*+ ,%-./)(*+ 01'231'*+
!"#! $%&'(&%)* %+%,-"%&!.*" %,(/.%" $&*$&-!0%0!" 0%" "!1/2+3-%" !. R.
!"#"$%& '('() !"# un $%# &!'$()*+# ,! -#./0!& 0!#+&1 2+3!%/& '$! un 4
• )5/6,!*0!&*!)7! &! un+1 ≥ un 8#0# 7/,/ n ∈ N∗;
• *0!&*!)7! &! un+1 > un 8#0# 7/,/ n ∈ N∗;
• )5/6*0!&*!)7! &! un ≥ un+1 8#0# 7/,/ n ∈ N∗;
• ,!*0!&*!)7! &! un > un+1 8#0# 7/,/ n ∈ N∗.
!"#"$%& '('(* !"# un $%# &!'$()*+# ,! -#./0!& 0!#+&1 9)75/ un 4 ,!)/%+)#,# %/):6
7/)# &! 8!07!)*!0 # $% ,/& 7+8/& ,!&*0+7/& )# 2!;)+<5/ =1=1>1
!"#$% &'&'( ?/&70! '$! # &!'$()*+# un = n+1n2+2
4 %/):7/)#1
!"#$%&"' 4!5!.*" .*"#&%& 1/! un $!&#!+3! % /. 0*" #-$*" 0!"3&-#*" +% 4!6+-78* 9:9:;:
<!.*" 1/! un = n+1n2+2
! un+1 =(n+1)+1(n+1)2+2
= n+2n2+2n+3
. =!&-63%&!.*" "! un+1 ≤ un
n+ 2
n2 + 2n+ 3≤ n+ 1
n2 + 2
⇔ (n2 + 2)(n+ 2) ≤ (n+ 1)(n2 + 2n+ 3)
⇔ n3 + 2n2 + 2n+ 4 ≤ n3 + 3n2 + 5n+ 3
⇔ 1 ≤ n2 + 3n.
> ?,#-.% 0!"-(/%,0%0! @ 5!&0%0!-&% $%&% #*0* n. A*(*B un = n+1n2+2
@ 0!3&!"3!+#! !B %""-.B
.*+C#*+%:
!"#"$%& '('(+ !"#% un $%# &!'$()*+# )$%40+*#@ C ! K ,/+& )A%!0/& 0!#+&1 2+3!%/&
'$! C 4 .+%+7#)7! +)B!0+/0 ,! un &! C ≤ un 8#0# 7/,/ n ! '$! K 4 .+%+7#)7! &$8!0+/0 ,! un
&! K ≥ un 8#0# 7/,/ n.
!"#$% &'&'& C/)&+,!0!%/& # &!'$()*+# %/):7/)# ,!*0!&*!)7! un = n+1n2+2
*$"/& 7!0%/& &5/
23, 36, 411, 518, · · · ! *$"/ .+%+7! 4 L = 0. 9)75/@ 7/,/ )A%!0/ 0!#. C ≤ 0 4 .+%+7#)7! +)B!0+/0 ,!
un ! 7/,/ K ≥ 234 .+%+7#)7! &$8!0+/0 ,! un, 8/+& un < u1 =
23.
!"#"$%& '('(, !"# un $%# &!'$()*+# )$%40+*# '$! 8/&&$+ .+%+7#)7!& +)B!0+/0!& ! &$8!6
0+/0!&@ !)75/ un 4 ,+7# &!'$()*+# .+%+7#,#1
!)*!+,-./% &'&'0 D/7! '$! $%# &!'$()*+#@ 8#0# &!0 .+%+7#,#@ )5/ 80!*+&# 7!0 .+%+7!1 E/0
!F!%8./@ un = (−1)n )5/ 7!% .+%+7!@ %#& 4 .+%+7#,#1
!" #$ %&%&' G/,# &!'$()*+# %/):7/)# .+%+7#,# !% R 4 */)-!0H!)7!1
!" #$ %&%&( !"#% un ! yn &!'$()*+#& )$%40+*#& !% R 7#+& '$! limn→∞
un = a !
limn→∞
yn = b. 9)75/ &5/ -I.+,#& #& #;0%#<J!&K
()* limn→∞
c = c;
;DE
!!" limn→∞
cun = ca;
!!!" limn→∞
(un ± yn) = a± b;
!#" limn→∞
unyn = ab;
#" ! b 6= 0 ! yn 6= 0 !"#$% limn→∞
un
yn= a
b;
#!" limn→∞
cnk = 0, &! k ' ()* +%"&#*"#! ,%&-#-.*/
!" #$%&'( )*+$%&,-(
$ !"#"$%& '()(* !0* un : N∗ → R ()* &!1(2"+-* "()'3-+*/ 4!"%)-"*)%& %&'!( -"5"-#*
6 &%)* 7! #%7%& %& -"5"-#%& #!3)%& 7!&&* &!1(2"+-*8 %( &!0*8 ()* &'3-! ' ()* !9,3!&&$% 7*
:%3)*
∞∑
n=1
un = u1 + u2 + u3 + · · ·+ uk + · · · .
; &!1(2"+-* un, +(0%& -"5"-#%& #!3)%& &$% &%)*7%&8 ' +<*)*7* 7! #!3)% =!3*> %( n−'&-)%#!3)% 7* &'3-!/
!"#$%"# &"'$()")$"# )* "#$!+* +" #,'("# #-*. /*0* #" +"$"'0()1 * '"#!2$1+* +" !01
#*01 ()3)($14 5*+1 #,'(" &*##!( !01 #*01 3)($14
61##1'"0*# 1 '"#&*)+"' $1(# 7!"#$%"# )* +"#")8*28(0")$* +* '"#$1)$" +"#$" 91&:$!2*; <*
")$1)$*= "#$1'"0*# 0!($* 01(# &'"*9!&1+*# 9*0 * >1$* +" +"$"'0()1' #" !01 #,'(" ()3)($1
&*##!( *! )-* !01 #*01 3)($1 +* 7!" &'*&'(10")$" ")9*)$'1' * 812*' +"#$1 #*01;
/*0"?1'"0*# 9*0 * 9*)9"($* +" %)*+% ,+'-!+!% +" !01 #,'(";
$ !"#"$%& '()(+ !0*
∞∑
n=1
un ()* &'3-!/ ; &%)* 7%& ,3-)!-3%& k #!3)%& 7!&#* &'3-!8 7*7*
,%3
Sk =k
∑
n=1
un = u1 + u2 + u3 + · · ·+ uk
' 7!"%)-"*7* &%)* ,*3+-*> 7* &'3-! 7*7*/
<*$" 7!" 1# #*01#
S1 = u1
S2 = u1 + u2 = S1 + u2
S3 = u1 + u2 + u3 = S2 + u3
· · ·Sk = Sk−1 + uk
>*'010 !01 #"7!@)9(1= 9A101+1 +" %(./01-!+ 2( %)*+% ,+'-!+!%; B" "#$1 #"7!@)9(1
9*)8"'C('= *! #"D1= #" "E(#$(' S $12 7!" limk→∞
Sk = S, +(F"0*# 7!" 1 #,'(" +1+1 -)1#('3( &1'1
S " +")*$1'"0*#
∞∑
n=1
un = S.
B" )-* "E(#$(' $12 S, +('"0*# 7!" 1 #,'(" 2!#('3(= #(C)(391)+* 7!" )-* &*+"0*# *G$"'
!0 812*' 3)($* &1'1 1 #*01 +1# ()3)($1# &1'9"21# +1 #,'(";
61'1 0"2A*' ")$")+(0")$*= 810*# 9*)#(+"'1' " 1)12(#1' !0 "E"0&2*;
HIJ
!"#$% &'(') !"#$%& ' %&()' *!& )&"(#$&+&" $# !$,-&".,/#/&0 !( &.%!/#$%& /# 1/&.+
/&-&"2 "&+&3&" !(# (&.#/# /& .&! )#,0 &( !$,/#/&. ('$&%2",#.0 *!& '3/&/&+& 4 .&*!5$+,#
un =20000
n(n+ 1), '$/& n +'""&.)'$/& #' $6(&"' /# )#"+&7# # .&" "&+&3,/#8 9&":!$%#;.&
<,= >!#7 ' ('$%#$%& *!& ' &.%!/#$%& /&-&"2 "&+&3&" #%? ' @$#7 /# A#+!7/#/&0 .!)'$/' *!& &7&
+'$+7!# ' +!".' &( BC (&.&.D
<,,= E' +#.' /' &.%!/#$%& )&"(#$&+&" $# !$,-&".,/#/& ,$/&@$,/#(&$%&0 +'(' @+#"2 ' ('$;
%#$%& "&+&3,/'D
!"#$%!& ! "#$%&'#! (&)!#*! $&%&+*,#! "&'- &!./,#).& !0- ,#,#! "&'# !&1/2)%*# 1/& ,&!3
%$&4& - 4#'-$ ,# (&!#,#5 1/& !0-
10000,10000
3,
5000
3, 1000,
2000
3,
10000
21,
2500
7, · · ·
6#$# $&!"-),&$ # "$*(&*$# "&$7/).#5 4#(-! &!%$&4&$ - "$-+'&(# )- 8-$(#.- ,& /(# !9$*&
*):)*.#5 *!.- 95
∞∑
n=1
20000
n(n+ 1)= 10000 +
10000
3+
5000
3+ 1000 +
2000
3+
10000
21+
2500
7+ · · ·
;! "$*(&*$-! .&$(-! ,#! !-(#! "#$%*#*! ,&!.# !9$*& !0- ,#,#! "-$
S1 = u1 = 10000,
S2 = S1 + u2 =40000
3,
S3 = S2 + u3 = 15000,
S4 = S3 + u4 = 16000
7-$#5 "$&%*!#(-! ,&.&$(*)#$ /(# &<"$&!!0- "#$# - .&$(- 7&$#' ,&!.# !-(#= 6#$# *!!-5
$&&!%$&4&(-! - .&$(- 7&$#' ,# !9$*& /!#),- ,&%-("-!*>0- &( 8$#>?&! "#$%*#*!5 .-(#),-
20000
n(n+ 1)=
A
n+
B
n+ 1=
A (n+ 1) +Bn
n(n+ 1)=
A+ (A+ B)n
n(n+ 1)
& -+.&),- 1/&
{
A = 20000A+B = 0
⇒ A = 20000 & B = −20000.
@&!!& (-,- # !9$*& ,#,# "-,& !&$ $&&!%$*.# %-(-
∞∑
n=1
20000
n(n+ 1)=
∞∑
n=1
(
20000
n− 20000
n+ 1
)
& # !-(# ,-! !&/! k−"$*(&*$-! .&$(-! 9 ,#,# "-$
Sk =
(
20000− 20000
2
)
+
(
20000
2− 20000
3
)
+ · · ·+(
20000
k− 20000
k + 1
)
& %-(- "-,&(-! !*("'*:%#$ #'7/)! .&$(-! *).&$(&,*A$*-!5 -+.&(-! 1/&
Sk = 20000− 20000
k + 1,
BCD
! "#$%&
Sk =20000k
k + 1.
' (#)* + , -#+. /#+)01%+ 2!# %" " 3%" ,%+1)%)" -#*#+3)4%-%" %4*#+) +3#4*# 1 ++#", 4-#3
5" 6 +4#1)-%" , + #"*% #7,+#""8 9
: 3 % " (!;8 ,%+% % 2!#"*8 <)= - #7#3,( 1 ++#", 4-# 5 "#7%>?")3% " 3%& *#3 "
2!#
S60 =20000 · 60
61= 19672.
@#""# 3 - & %,A" BC 3#"#"& #"*!-%4*# *#+. +#1#D)- !3 3 4*%4*# -# 19672 !4)-%-#"
3 4#*.+)%"9
E%""%+#3 " %> +% % +#", 4-#+ % "#>!4-% 2!#"*8 9 F% G)>!+% H9I , -#3 " /#+ 1 3, +J
*%3#4* ,%+% 1+#"1)3#4* -% " 3% -% "?+)#9
Sk
k
G)>!+% H9IK L"*)3%*)/% ,%+% 1+#"1)3#4* -% "?+)#
E +*%4* & "# #"*!-%4*# 01%+ )4-#04)-%3#4*# 4% !4)/#+")-%-#& D"#+/%4- >+.01 &
, -#3 " %0+3%+ 2!# 48 +#1#D#+)% 3%)" - 2!# 20000 !4)-%-#" 3 4#*.+)%"9 M"" ")>4)01%
2!# % " 3% -% "?+)# *#3 ()3)*# 20000 2!%4- % 2!%4*)-%-# -# ,%+1#(%" *#4-# ,%+% )404)* &
! "#$%&
limk→∞
Sk = limk→∞
20000k
k + 1= 20000.
L3 !*+%" ,%(%/+%"& % "?+)# 1 4/#+># ,%+% 20000 # , -#3 " #"1+#/#+
∞∑
n=1
20000
n(n+ 1)= 20000.
: 3 /)3 " %1)3%& % " 3% -# !3% "?+)# )404)*% ? D*)-% ,#( ()3)*# -% "!% "#2!N41)% -#
" 3%" ,%+1)%)"9 O"")3& -#04)3 " ()3)*# -# !3% "?+)# - 3#"3 3 - 1 3 2!# 6 ) -#04)-
()3)*# -# !3% "#2!N41)%9
!"!# $%&' () *&' $+,-)
!"#"$%& '()(' !"#
∞∑
n=1
un $%# &'()! *$"# &!+$,-*)# .! &/%#& 0#(*)#)& ' Sk. 1)2!%/&
+$! / -3%!(/ S ' # &/%# .# &'()!4 .!-/5#-./ S =∞∑
n=1
un, &! S 6/( / 7)%)5! .! Sk +$#-./ k
5!-.!( 0#(# / )-8-)5/4 /$ &!"#4 &! .#./ ε > 0 0$.!(%/& !-*/-5(#( N0 > 0 5#7 +$!4 0#(# 5/./
k > N0 9#7! # .!&):$#7.#.! |Sk − S| < ε.
!!
!"#$% &'('( !"#$%&'& ( #)'$& !*+$%( "! ,-&./0! 123245 %(%( /!'
∞∑
n=1
20000
n(n+ 1). 6!#+'&
78&
∞∑
n=1
20000
n(n+ 1)= 20000.
!"#$%!& !"! #$"!% &'$"&( & %)*+,-'$& .) %!"&% /&0'$&$% .& %10$) .&.& 1 Sk = 20000kk+1
.
2)#)"!% )-34! "!%30&0 *+) limk→∞
20000kk+1
= 20000, !+ %)5&( *+) .&.! ε > 0 /!.)"!% )-'!-30&0
N0 > 0 3&6 *+) /&0&( %) k > N0 )-34! |Sk − 20000| < ε. !"!
|Sk − 20000| =∣
∣
∣
∣
20000k
k + 1− 20000
∣
∣
∣
∣
=
∣
∣
∣
∣
20000k − 20000k − 20000
k + 1
∣
∣
∣
∣
=
∣
∣
∣
∣
−20000k + 1
∣
∣
∣
∣
3)"!% *+) & .)%$7+&6.&.) .)%)5&.& %)08 #86$.& %)
20000
k + 1< ε ⇒ 20000 < kε+ ε ⇒ 20000− ε
ε< k.
!-%)*+)-3)")-3)( /!.)"!% 3!"&0 N0 =20000− ε
ε) & 2)9-$:4! ;<=<> )%3&08 %&3$%?)$3&<
@+/!-A&"!% *+) %) .)%)5& %&B)0 & /&03$0 .) *+&6 /&0')6& & .$?)0)-:& )-30) ! "!-3&-3)
) ! 3!3&6 & 0)')B)0 %)08 ")-!0 .! *+) 300 +<"<< C&0& !B3)0 & 0)%/!%3& 3!"&"!% ε = 300 )
!B3)0)"!% N0 =20000− 300
300= 65, 667. D%%! %$7-$9'& *+) )" 3!.&% &% /&0')6&%( & /&03$0 .&
%)E&71%$"& %)E3&( & .$?)0)-:& )-30) ! "!-3&-3) ) ! 6$"$3) 1 ")-!0 .! *+) 300 +<"<<@+/!-A&"!% *+) %) .)%)5& %&B)0 & /&03$0 .) *+&6 /&0')6& & .$?)0)-:& )-30) ! "!-3&-3)
) ! 6$"$3) 1 ")-!0 .! *+) 200 +<"<< C&0& !B3)0 & 0)%/!%3& 3!"&"!% ε = 200 ) !B3)0)"!%
N0 =20000− 200
200= 99. D%%! %$7-$9'& *+) )" 3!.&% &% /&0')6&%( & /&03$0 .& /&0')6& .)
-F")0! 99( & .$?)0)-:& )-30) ! "!-3&-3) ) ! 6$"$3) 1 ")-!0 .! *+) >GG +<"<<
!"!# $%&'() *+,-(&.(,/()
' !"#"$%& '()(* 9&:(
∞∑
n=1
un 8.( #)'$& & #&:( Sk ( #!.( /(';$(0 %!# +&'.!# %&##( #)'$&2
<$=&.!# 78&
∞∑
n=1
un ) (!)*+,-+).+ #& limk→∞
Sk &-$#+&2 (#! ;!"+'>'$!5 %$=&.!# 78& ( #)'$& )
/0*+,-+).+2
!"#$% &'(') ? #)'$&
∞∑
n=1
20000n(n+1)
%! ,-&./0! 12324 ) ;!"@&'A&"+& /!$#
limk→∞
Sk = limn→∞
20000k
k + 1= 20000.
!"#$% &'('*+ <&+&'.$"& #& ( #)'$&
∞∑
n=1
2n
5n−1) ;!"@&'A&"+& !8 %$@&'A&"+&2
!"#$%!& 2)#)"!% #)0$9'&0 %) & %)*+,-'$& .) %!"&% /&0'$&$% .)%3& %10$) 3)" 6$"$3)< H!.&%
&% %10$)% *+) &/0)%)-3&" )%%) "!.)6! I%10$)% 7)!"130$'&%J /!.)" %)0 0)%!6#$.&% '!-?!0") !
"!.)6! *+) %)7+)<
I$J K%'0)#)"!% & %!"& .!% k /0$")$0!% 3)0"!%L
Sk = 2 +22
5+
23
52+
24
53+ · · ·+ 2k
5k−1
>MN
!!" #$%&!'%!()*+, Sk '+-
25
2
5Sk =
22
5+
23
52+
24
53+ · · ·+ 2k
5k−1+
2k+1
5k
!!!" .+*)*+, ) /!01-123) 12&-1 +, -1,$%&)/+, /1 !" 1 !!"4 +5&12/+
Sk −2
5Sk =
(
2 +22
5+
23
52+ · · ·+ 2k
5k−1
)
−(
22
5+
23
52+ · · ·+ 2k
5k−1+
2k+1
5k
)
+$ ,16)4
3
5Sk = 2− 2k+1
5k
+$ )!2/)4
Sk =10
3− 5
3
2k+1
5k=
10
3− 10
3
(
2
5
)k
1 (+*+
2
5< 1, &1*+, 7$1 )
S = limk→∞
Sk = limk→∞
10
3− 10
3
(
2
5
)k
=10
3.
8+2,17$12&1*12&14 ) ,9-!1
∞∑
n=1
2n
5n−1(+2:1-;1 ')-)
10
3.
!"#$% &'(')) !"#!$%& # $&%'# (&%)* +) ,&-./!"0) +& ,#'), 1)%"0)0, +) ,2%0&
∞∑
n=1
−4(2n+ 3)(2n− 1)
.
3 ,&(.0%4 +&$&%'0!& ,& ) ,2%0& "#!5&%(& #. +05&%(&4 #6$&!+# # 5)*#% +& ,.) ,#')4 ,& 1#,,75&*8
!"#$%!& <+&1 7$1
∞∑
n=1
−4(2n+ 3)(2n− 1)
=1
2n+ 3− 1
2n− 1, ),,!* &1*+, 7$1
∞∑
n=1
−4(2n+ 3)(2n− 1)
=∞∑
n=1
(
1
2n+ 3− 1
2n− 1
)
.
=+;+4 ) ,17$>2(!) /), ,+*), ')-(!)!, 9?
Sk =k
∑
n=1
(
1
2n+ 3− 1
2n− 1
)
=
(
1
5− 1
)
+
(
1
7− 1
3
)
+
(
1
9− 1
5
)
+
(
1
11− 1
7
)
+ · · ·+
+ · · ·+(
1
2k − 1− 1
2k − 5
)
+
(
1
2k + 1− 1
2k − 3
)
+
(
1
2k + 3− 1
2k − 1
)
= −1− 1
3+
1
2k + 1+
1
2k + 3
@+-&)2&+4 + &1-*+ ;1-)% /) ,17$>2(!) /1 ,+*), ')-(!)!, /) ,9-!1 /)/) 9 Sk = −4
3+
1
2k + 1+
1
2k + 3.
ABC
!" #$%&'()! * +,"'$ -!&.$"/$ +$ limk→∞
Sk $0'+1$ $ * +!2* #* +,"'$ , ! .*3!" #! 3'2'1$4
5!2!
limk→∞
Sk = limk→∞
(
−4
3+
1
2k + 1+
1
2k + 3
)
= −4
3.
6 +,"'$ #*#* -!&.$"/$ $ +7* +!2* , S = −43.
!"#$%&'(#")
84 92* #*+ :"!:"'$#*#$+ #*+ +,"'$+ '&%&'1*+ , ;7$ * -!&.$"/<&-'* !7 #'.$"/<&-'* &)!
, *=$1*#* +$ +7>1"*'"2!+ !7 *#'-'!&*"2!+ 72 &?2$"! %&'1! #$ 1$"2!+ * $3*+4 !"
$0$2:3!@ +$ &! A0$2:3! B4C4D ! $+17#*&1$ +E -!2$(*++$ * "$-$>$" * :"'2$'"* :*"-$3*
*:E+ B 2$+$+@ * +,"'$ +$"'* $+-"'1* -!2 n = 6 &! :"'2$'"! 1$"2!@ !7 +$F*@∞∑
n=6
20000
n(n+ 1),
$ * +!2* +$"'* S = 20000− S5. G$ :!" !71"! 3*#!@ ! +$7 :*' #$-'#'++$ &!+ :"'2$'"!+ 8H
2$+$+ #*" 72* 2$+*#* %0* #$ IHHH7424 :!" 2<+ $ '&'-'*" ! :*/*2$&1! -!2 n = 1 &!
#,-'2! :"'2$'"! 2<+@ * +!2* +$"'* S = 2000(10) + limk→∞
20000k
k + 1. A2 *2>!+ !+ -*+!+ *
+,"'$ -!&1'&7*"J -!&.$"/$&1$4
I4 G$ * +,"'$
∞∑
n=1
un , -!&.$"/$&1$ $ * +,"'$
∞∑
n=1
yn , #'.$"/$&1$@ $&1)! * +,"'$∞∑
n=1
(un+ yn) ,
#'.$"/$&1$4 K! $&1*&1!@ +$ *+ +,"'$+
∞∑
n=1
un $
∞∑
n=1
yn +)! #'.$"/$&1$+@ * +,"'$∞∑
n=1
(un+ yn)
:!#$ +$" -!&.$"/$&1$ !7 #'.$"/$&1$4
D4 G$
∞∑
n=1
un , 72* +,"'$ -!&.$"/$&1$ #$ 1$"2!+ :!+'1'.!+@ +$7+ 1$"2!+ :!#$2 +$" "$*/"7L
:*#!+ #$ ;7*3;7$" 2!#! $ * +,"'$ "$+731*&1$ 1*2>,2 +$"J -!&.$"/$&1$ $ 1$"J * 2$+2*
+!2* ;7$ * +,"'$ #*#*4
!" #$ %&'&() !"#
∞∑
n=1
un $%# &'()! ! α ∈ N∗. ! # &'()!
∞∑
n=α
un = uα + uα+1 + uα+2 + · · ·
*+( ,+-.!(/!-0!1 !-02+ # &'()!
∞∑
n=1
un = u1 + u2 + u3 + · · ·+ uk + · · ·
0#%3'% &!(4 ,+-.!(/!-0!5
* !"#$%&'()"* G7:!&#! ;7$ * +,"'$
∞∑
n=α
un , -!&.$"/$&1$@ 1$2!+ ;7$ $3* :!++7' 72* +!2*4
G$F* Sk−α ! 1$"2! /$"*3 #* +$;7<&-'* #$ +7*+ +!2*+ :*"-'*'+@ 1*3 ;7$ S = limk→∞
Sk−α $ +$F*
Sα = u1 + u2 + u3 + · · ·+ uα. M$++$ 2!#!@ ! 1$"2! /$"*3 #* +!2* :*"-'*3 #* +,"'$∞∑
n=1
un +$"J
Sk = Sα+Sk−α $@ :!"1*&1!@ limk→∞
Sk = limk→∞
Sα+ limk→∞
Sk−α, #!&#$ +$/7$ ;7$ limk→∞
Sk = Sα+S.
5!&+$;7$&1$2$&1$@
∞∑
n=1
un , -!&.$"/$&1$4
8NH
!"#!$%&'&%(
!"#$
∞∑
n=1
un = u1 + u2 + u3 + · · ·+ uk + · · ·
!
∞∑
n=1
yn = y1 + y2 + y3 + · · ·+ yk + · · ·
%&#' '()*!' +&! ,-./!)0!$ 1#)# S ! S ′, )!'1!,2*/#$!.2!3 !.24- '4- /56*%#' #' '!0&*.2!'
1)-1)*!%#%!'7
8*9
∞∑
n=1
kun = k∞∑
n=1
un 1#)# 2-%- k ∈ R, -& '!"#3 # '()*!
∞∑
n=1
kun ,-./!)0! 1#)# kS.
8**9
∞∑
n=1
(un ± yn) =∞∑
n=1
un ±∞∑
n=1
yn, -& '!"#3 # '()*!
∞∑
n=1
(un ± yn) ,-./!)0! 1#)# S + S ′.
!" #$%&'()$ %*+*,,-.'/ 0/./ #$%1*.23%+'/
:4- !;*'2! &$# )!0)# 0!)#6 1#)# /!)*<,#) '! &$# '()*! ( ,-./!)0!.2! -& .4-7 =-$- /!)!$-'
.-' 1)>;*$-' *2!.'3 ?5 ,)*2()*-' +&! %4- )!'1-'2#' # 2*1-' 1#)2*,&6#)!' %! '()*!'7 @-)($3
/!)*<,#.%- '! &$# '()*! .4- 1-''&* # ,-.%*A4- .!,!''5)*# 1#)# ,-./!)0B.,*#3 '#C!)!$-' +&!
!6# .4- ( ,-./!)0!.2!7 D''# ,-.%*A4-3 ( %#%# 1!6- 2!-)!$# #C#*;-7
!" #$ %&'&( !
∞∑
n=1
un " #$% &"'(! )*+,!'-!+.!/ !+.0* limn→∞
un = 0.
) !"#$%&'()"* &1-.?#$-' +&! # '()*!
∞∑
n=1
un ,-./!)0! 1#)# S, !.24- 1-%!$-' #<)$#)
+&! limk→∞
Sk = S, %! $-%- +&!3 1!6# E!<.*A4- F7G7H3 %#%- ε > 0 1-%!$-' !.,-.2)#) N0 > 0
2#6 +&! 1#)# 2-%- k > N0 /#6! # %!'*0%#%! |Sk − S| < ε2
! |Sk−1 − S| < ε2. =-$-
Sk = Sk−1 + uk, 2!$-' +&! uk = Sk − Sk−1 ! #''*$3
|uk − 0| = |Sk − Sk−1 − 0|= |Sk − S + S − Sk−1|= |(Sk − S) + (S − Sk−1)|= |Sk − S|+ |S − Sk−1|≤ |Sk − S|+ |Sk−1 − S|<
ε
2+
ε
2= ε.
I''*$3 1!6# E!<.*A4- F7J7K3 '!0&! +&! limk→∞
uk = 0.
L$# ,-.'!+&B.,*# $&*2- *$1-)2#.2! %!''! 2!-)!$# ( - ,-)-65)*- # '!0&*)7
) ! "#!$ %&'&( !1%
∞∑
n=1
un #$% &"'(! .%2 3#! limn→∞
un 6= 0, !+.0*∞∑
n=1
un " 4(,!'-!+.!5
+ !,-" ./0/1 6 &"'(!
∞∑
n=1
2n+23n+5
" 4(,!'-!+.! 17 3#! limn→∞
un = limn→∞
2n+23n+5
= 236= 0.
MHM
!"#$% &'(') !"#$%
∞∑
n=1
1n" &'( )*% lim
n→∞un = lim
n→∞
1n
= 0+ $!&, "+ -,!!*$ ' .,/0$12,
/%.%!!3#$' -'#' .,/4%#56/.$'7 8, %/&'/&,+ /2, -,0%9,!+ !%9 '-($.'# ,*&#,! &%!&%! 0% .,/:
4%#56/.$'+ ';#9'# !% %(' " .,/4%#5%/&% ,* 0$4%#5%/&%7
!*+!,-./0% &'('& <,#&'/&, ;)*%9 '&%/&,!+ !% , limn→∞
un 6= 0 -#,4':!% )*% ' !"#$% " 0$4%#:
5%/&%7 ='!+ !% limn→∞
un = 0 ' !"#$% -,0% .,/4%#5$# ,* 0$4%#5$#+ -'#' $!!, /%.%!!$&'9,! %!&*0'#
.#$&"#$,! -'#' >'?%# &'( 4%#$;.'12,7
!"!#$%& '( %!)*+',-(& (./*'% "!%*.0(1$% )*! 2!"#-0!# 3!"-4,(" %! *#( %5"-! 5 ,$'3!"6
/!'0! $* 1-3!"/!'0!
!" #$%&'( )(*'+&,&(
!"!# $%&'( )*&+,-'.*
!"#"$%& '()(* !"#$%
∞∑
n=1
1
n" 0%/,9$/'0' !"#$% @'#9A/$.'7
7 %5"-! 8("#9'-,( 5 *#( 1(% %5"-!% #(-% -#2$"0('0!% 1( #(0!#:0-,(; <!* '$#! %*"/!
!# ,$'!=>$ ,$# $% %$'% 8("#9'-,$% 2"$1*?-1$% 2!.( 3-@"(A>$ 1! *#( ,$"1( #*%-,(.;
7 %5"-! 8("#9'-,(& !#@$"( 2$%%*( ( ,$'1-A>$ '!,!%%:"-( 2("( ,$'3!"/+',-(& 5 *#( %5"-!
1-3!"/!'0!; 7 1-3!"/+',-( 1( %5"-! 8("#9'-,( '>$ 5 0"-3-(.; <*( .!'0( 1-3!"/+',-( %! 0$"'(":
!3-1!'0! )*('1$ !=(#-'("#$% %*(% %$#(% 2(",-(-% ,$# #(-$" 1!0(.8!; B( 3!"1(1!& 3(#$%
#$%0"(" )*! ( %!)*+',-( 1! %$#(% 2(",-(-% Sn 1( %5"-! 8("#9'-,( '>$ ,$'3!"/!& 2$-% (1#-0!
%*@%!)*+',-(% 1-3!"/!'0!%; C("( -%%$& 3(#$% ,$'%-1!"(" (% %$#(% S2, S4, S8, S16, S32, · · · ,*D$%E'1-,!% %>$ %!#2"! 2$0+',-(% 1! 2, F$"#('1$ ( %*@%!)*+',-( S2n 1! Sn. G!#$% )*!
S21 = S2 = 1 +1
2>
1
2+
1
2=
2
2
S22 = S4 = S2 +1
3+
1
4> S2 +
(
1
4+
1
4
)
= S2 +1
2>
3
2
S23 = S8 = S4 +1
5+
1
6+
1
7+
1
8> S4 +
(
1
8+
1
8+
1
8+
1
8
)
= S4 +1
2>
4
2
S24 = S16 = S8 +1
9+
1
10+
1
11+
1
12+
1
13+
1
14+
1
15+
1
16
> S8 +
(
1
16+
1
16+
1
16+
1
16+
1
16+
1
16+
1
16+
1
16
)
= S8 +1
2>
5
2
! (%%-# %*,!%%-3(#!'0!& 1! F$"#( )*! 2$1!#$% -'0*-" )*! S2n >n+ 1
22("( 0$1$ n ∈ N∗.
H!%0( F$"#(& 0!#$% )*!
limn→∞
S2n ≥ limn→∞
n+ 1
2=∞,
$ )*! '$% 1-? )*! S2n 5 *#( %*@%!)*+',-( 1-3!"/!'0! 1! Sn. I$# -%%$& 0!#$% )*! Sn 0(#@5#
1-3!"/!& 2$-% 1$ ,$'0":"-$ -"E(#$% ,$'0"("-(" $ G!$"!#( J;K;K; I$#$ ( %!)*+',-( 1! %$#(%
2(",-(-% 1( %5"-! 8("#9'-,( 1-3!"/!& ,$',.*E#$% )*! ( 2"L2"-( !"#$% &'#()*$+' ,$-%#.%;
!D(#$% (./*#(% %$#(% 2(",-(-% 1( %5"-! 8("#9'-,(& $@0-1(% ,$# (*=E.-$ 1$ M7CNO P&
)*! '$% #$%0"( ( F$"#( .!'0( ,$# ( )*(. ( %$#( 1( %5"-! 0!'1! ($ -'4'-0$;
S10 = 2, 9289 S100 = 5, 1873 S1000 = 7, 485Sum milhao = 14, 392 Sum bilhao = 21, 300 Sum trlhao = 28, 208.
QRS
!"!# $%&'( )(*+%,&'-.
!"#"$%& '()(* !"#$%"&$#' '()%! *!#$(+)%,& - +#.& '()%! .& /#)$&
∞∑
n=1
a1qn−1, #".! q
( .!"#$%"&.& )&01#2
!"#$% &'('& 3",#"+)! & '#$& .& '()%! *!#$(+)%,& ! !'+4.! '4& ,#"5!)*6",%&2
!"#$%&"' !"#$%&'&(!# ) #*'$& +&!(*,'$-)
∞∑
n=1
a1qn−1 = a1 + a1q + aq2 + · · ·+ a1q
n−1 + · · ·
& ) #!() %!# #&.# n−/'$(&$'!# ,&'(!#0 %)%) /!'
Sn = a1 + a1q + aq2 + · · ·+ a1qn−1.
1.2,$/2$-)"%! )(3!# !# 2)%!# %&##) $+.)2%)%& /&2) ')45! q !3,&(!#
qSn = a1q + a1q2 + a1q
3 + · · ·+ a1qn
& ,!()"%! ) %$6&'&"7) &",'& )# %.)# 82,$()# &9/'&##:� !3,&(!#
qSn − Sn = (a1q + a1q2 + a1q
3 + · · ·+ a1qn)− (a1 + a1q + aq2 + · · ·+ a1q
n−1) ,
(q − 1)Sn = a1qn − a1 = a1(q
n − 1),
Sn =a1(q
n − 1)
(q − 1).
;)') &#,.%)' ) -!"<&'+="-$) %&##) #*'$& %&<&(!# -!"#$%&')' ,'=# -)#!#>
?@A B& q = 1 &",5! limn→∞
Sn = limn→∞
a1(qn − 1)
(q − 1)=∞ & ) #*'$& * %$<&'+&",&C B& q = −1 &",5!
Sn ,&( %!$# <)2!'&# /)') ! 2$($,& &0 /!',)",!0 ) #*'$& * %$<&'+&",&C
?@@A B& |q| > 1 &",5! limn→∞
Sn = limn→∞
a1(qn − 1)
(q − 1)=∞ & ) #*'$& * %$<&'+&",&C
?@@@A B& |q| < 1 &",5! limn→∞
Sn = limn→∞
a1(qn − 1)
(q − 1)= lim
n→∞
a1qn
q − 1+ lim
−a1(q − 1)
=−a1
(q − 1)& )
#*'$& * -!"<&'+&",&C
!"-2.#5!> ()* +,-./ 0/"),1-.2* , 3.4/-0/51/ +/ |q| ≥ 1 & * 2"54/-0/51/ +/
|q| < 1. D.)"%! |q| < 1 )$"%) ,&(!# E.&∞∑
n=1
a1qn−1 =
a1
1− q.
!"#$% &'(') 7 '()%!
∞∑
n=1
(
23
)n( ,#"5!)*!"+!8 9#%' '4& )&01# ( q = 2
3< 1. :; & '()%!
∞∑
n=1
(
32
)n( .%5!)*!"+! 9#%' '4& )&01# ( q = 3
2> 1.
FGH
!" #$%&'$%() *+ #(,-+$./,0%1 *+ 2'$%+)
!"#$% &%#'(&()%* % +(,)% -(,". $" *%)" $( !)" */,0(1 / 23&0. 2"4(, " 5(,06&"78% $"
&%#5(,-9#&0": ;%$()%* 5(,06&", *( !)" */,0( &%#5(,-( !*"#$% &,0+/,0%* <"," &%#5(,-9#&0"
=!( <"**",()%* " (*+!$", " *(-!0,:
!"!# $%&'(%&) *+ &,'-.%+/
0 !" #$ %&'&( !"#
∞∑
n=1
un $%# &'()! *#+ ,$! un+1 ≤ un -#(# *./. n ∈ N∗. !"# f (x)
$%# 0$123. -.&)*)4#5 6.1*71$# ! /!6(!&6!1*! 1. )1*!(4#+. [1,∞) *#+ ,$! f (n) = un -#(# *./.
n ∈ N∗. 81*3.5 &! # )1*!9(#+
∫ ∞
1
f (x) dx 6.14!(9)(5 # &'()!
∞∑
n=1
un *#%:'% &!(; 6.14!(9!1*!<
! # )1*!9(#+ /)4!(9)(5 # &'()! *#%:'% &!(; /)4!(9!1*!<
> $()%#*+,"78% $(*+( +(%,()" <%$(,3 *(, (*+!$"$" () =!".=!(, !) $%* .05,%* &%#*+"#+(*
#" ?0?.0%-,"6":
!"#$% &'(') =!()>,$! #& ?)-@*!&!& /. *!&*! /# )1*!9(#+ ! $*)+)A!B.5 &! -.&&74!+5 -#(# #1#+)&#(
# 6.14!(9C16)# /# &'()!
∞∑
n=1
ne−n.
!"#$%!& @%#*0$(,( " 2!#78% f(x) = xe−x, %?50")(#+( f(x) / &%#+A#!" ( <%*0+05" <","x ≥ 1. B".+" 5(,06&", =!( / $(&,(*&(#+(: C*"#$% % +(*+( $" <,0)(0," $(,05"$" +()%* =!(f ′(x) = e−x(1 − x) ( f ′(x) < 0 <"," +%$% x > 1, () x = 1 2!#78% "<,(*(#+" !) )3D0)%
.%&".1 (#+8% f(x) / $(&,(*&(#+( <"," +%$% x ≥ 1. @%)% "* '0<E+(*(* $% +(*+( $" 0#+(-,". (*+8%
5(,06&"$"* <%$()%* !+0.043F.% <"," (*+!$", " &%#5(,-9#&0" $" */,0(
∞∑
n=1
ne−n.
G +(*+( $" 0#+(-,". "6,)" =!( " */,0(
∞∑
n=1
ne−n &%#5(,-( *(1 " 0#+(-,". I =
∫ ∞
1
xe−xdx
&%#5(,-( ( " */,0( $05(,-( *( " 0#+(-,". $05(,-0,:
>**0)1
I =
∫ ∞
1
xe−xdx = limb→+∞
∫ b
1
xe−xdx
= limb→+∞
−xe−x∣
∣
∣
∣
∣
b
1
+
∫ b
1
e−xdx
= limb→+∞
(
−be−b + e−1 − e−b + e−1)
=2
e+ lim
b→+∞
(
− b
eb− 1
eb
)
=2
e.
@%)% " 0#+(-,". 0)<,E<,0" &%#5(,-(1 <(.% +(*+( $" 0#+(-,". " */,0(
∞∑
n=1
ne−n +")?/) &%#5(,-(:
!"!1 2(%&- 3 )4 2(%&- 5&3-%67+%89,&:+
' !"#"$%& '()(' D!1.%)1#%.& &'()! p *./#& #& &'()!& !&6()*#& 1# 0.(%#
∞∑
n=1
1
np, .1/! p '
$%# 6.1&*#1*! -.&)*)4#<
HIJ
!"#$ %&'(')!* # +,#*,"! -./.0 1!*! ,$&%2!* ! 3#45,*6743'! 2! $8*', p.
!"#$% &'(') !"#$% & '()*%+,-)'.& $& !/+.%
∞∑
n=1
1
np= 1+
1
2p+
1
3p+
1
4p+ · · ·+ 1
np+ · · · .
!"#$%!& 9#4$'2,*!42# f (x) =1
xp, &,"#$ :%, f 8 1#$'&'5!; 3#4&<4%! , 2,3*,$3,4&,; $!&'$=
>!),42# !$ !$ 3#42'?@,$ 2# +,#*,"! -./.0; 2, "#2# :%, 1#2,"#$ &#"!* ! '4&,6*!(
∫ ∞
1
1
xpdx = lim
n→∞
∫ n
1
1
xpdx.
+,"#$ &*7$ 3!$#$ ! 3#4$'2,*!*A
B'C D, p = 1 &,*,"#$ :%,
∫ ∞
1
1
xdx = lim
n→∞
∫ n
1
1
xdx = lim
n→∞ln x
∣
∣
∣
∣
∣
n
1
= limn→∞
(lnn− ln 1) =∞.
9#4$,:%,4&,",4&,; :%!42# '(); ! $8*',
∞∑
n=1
1
np=
∞∑
n=1
1
n8 *+,-./-01-. E#&, :%, 4,$&,
3!$#; &,"#$ ! $8*', F!*"G4'3!.
B''C D, p < 1 &,*,"#$ :%, 1− p > 0 , !$$'"
∫ ∞
1
1
xpdx = lim
n→∞
∫ n
1
1
xpdx = lim
n→∞
x1−p
1− p
∣
∣
∣
∣
∣
n
1
= limn→∞
(
n1−p
1− p− 1
1− p
)
=∞.
9#4$,:%,4&,",4&,; $, '2); ! $8*',
∞∑
n=1
1
np8 *+,-./-01-.
B'''C D, p > 1 &,*,"#$ :%, 1− p < 0 , !$$'"
∫ ∞
1
1
xpdx = lim
n→∞
∫ n
1
1
xpdx = lim
n→∞
x1−p
1− p
∣
∣
∣
∣
∣
n
1
= limn→∞
(
n1−p
1− p− 1
1− p
)
=−11− p
.
9#4$,:%,4&,",4&,; $, '3) ! $8*',
∞∑
n=1
1
np8 4!0,-./-01-.
!"#$% &'('* 0! !/+.%! &1&.2( !3( %2%456(! $% !/+.%! p.
7&8
∞∑
n=1
1
n9'()*%+,%)"%9 5(.! / #4& !/+.%:5 '(4 p = 9 > 1.
718
∞∑
n=1
1√n$.*%+,%)"%9 5(.! / #4& !/+.%:5 '(4 p = 1
2< 1.
HI-
!"!# $%&'(%&) *+ ,)-.+%+/0)
1 !" #$ %&'&' !"#
∞∑
n=1
un !" #$%&' ' #'("
∞∑
n=1
yn !" #$%&' ) (" )*+,'%-.+)&" / '%'!*#
'#0 1"%2 '+03*4
5&6 7'
∞∑
n=1
un 8*% !" #$%&' )*+,'%-'+0' ' 0 ≤ yn ≤ un 9"%" 0*1* n, '+03* " #$%&'
∞∑
n=1
yn $
)*+,'%-'+0':
5&&6 7'
∞∑
n=1
un 8*% !" #$%&' 1&,'%-'+0' ' yn ≥ un ≥ 0 9"%" 0*1* n, '+03* " #$%&'
∞∑
n=1
yn $
1&,'%-'+0':
!"#$%&'()"* !" #$%&'
∞∑
n=1
un ('& )*+!$ ,-./$+0$.1$ $
∞∑
n=1
yn ('& )*+!$ 1&2 3($ 0 ≤ yn ≤
un 4&+& 1-5- n. 6-'-
∞∑
n=1
un * ('& )*+!$ ,-./$+0$.1$7 & )$3(8.,!& 5$ )(&) )-'&) 4&+,!&!) Sn
1$' 2!'!1$ L, 5$ '-5- 3($ u1 + u2 + u3 + · · ·+ uk + · · · < L. 6-'- 0 ≤ yn ≤ un 4&+& 1-5-
n, )$0($ 3($
0 ≤ y1 + y2 + y3 + · · ·+ yk + · · · ≤ u1 + u2 + u3 + · · ·+ uk + · · · < L.
6-.)$3($.1$'$.1$7 & )$3(8.,!& 5$ )-'&) 4&+,!&!) 5$
∞∑
n=1
yn * 2!'!1&5& $7 &2*' 5!))-7
'-.91-.&: ;-0-7 4$2- <$-+$'& =:=:> * ,-./$+0$.1$ $7 &))!'7 & )*+!$
∞∑
n=1
yn * ,-./$+0$.1$:
!!" #$%&'
∞∑
n=1
un ('& )*+!$ 5!/$+0$.1$ $ yn ≥ un ≥ 0 4&+& 1-5- n. 6-'-
∞∑
n=1
un * ('&
)*+!$ 5!/$+0$.1$ & )(& )$3(8.,!& 5$ )-'&) 4&+,!&!) Sn .?- 1$' 2!'!1$7 5$ '-5- 3($ 5&5- ('
.@'$+- L > 0, $A!)1$ K > 0 1&2 3($ u1 + u2 + u3 + · · · + uk + · · · > L 4&+& 1-5- n > K.
6-'- yn ≥ un 4&+& 1-5- n, )$0($ 3($
y1 + y2 + y3 + · · ·+ yk + · · · ≥ u1 + u2 + u3 + · · ·+ uk + · · · > L.
6-.)$3($.1$'$.1$7 & )$3(8.,!& 5$ )-'&) 4&+,!&!) y1 + y2 + y3 + · · · + yk + · · · .?- *
2!'!1&5& $7 &))!'7 & )*+!$
∞∑
n=1
yn * 5!/$+0$.1$:
!+ !,-" ./0/12 ;#"+1* * <'*%'!" =:>:> '#0 1' " )*+,'%-.+)&" 1" #$%&'
∞∑
n=1
n
n3 + n2 + n+ 1.
!"#$%!& 6-.B-+'$ - <$-+$'& =:C:C7 5$/$'-) $.,-.1+&+ ('& )*+!$ 3($ )&D$'-) )$+ ,-./$+E
0$.1$ -( 5!/$+0$.1$ $ B&F$+ & ,-'4&+&G?- 5- 1$+'- 0$+&2 5$))& )*+!$ ,-' & )*+!$ $' $)1(5-:
H' 4+-,$5!'$.1- ()&5- 4&+& $.,-.1+&+ (' 1$+'- 0$+&2 &5$3(&5- * '&%-+&+ - 1$+'- 0$+&2
5& )*+!$ 4+-4-)1&: I&'-) 5$),+$/$+ - 4+-,$))-:
!" <$'-) 5(&) B-+'&) 5$ '&%-+&+ (' 3(-,!$.1$J &('$.1&.5- - 5$.-'!.&5-+ -( 5!'!.E
(!.5- - 5$.-'!.&5-+: K- 1$+'- 0$+&2 5& )*+!$ $' $)1(5-7 /&'-) 5!'!.(!+ - 5$.-'!E
.&5-+ 4&))- & 4&))-
n
n3 + n2 + n+ 1<
n
n3 + n2 + n<
n
n3 + n2=
1
n(n+ 1).
L>M
! "#$%&'! ()*)+, -.%!/ 01$ 2 /34.$
∞∑
n=1
20000
n(n+ 1)3 5!6-$47$68$) 9!%! &!:$%!/ $/54$-$4
∞∑
n=1
20000
n(n+ 1)= 20000
∞∑
n=1
1
n(n+ 1), /$71$ ;&$'2 &4!&4.$:2:$ .<, 01$
∞∑
n=1
1
n(n+ 1)82%=3% 3
5!6-$47$68$)
;..< >2%!/ -$4.?524 01$, :$ @28!,
n
n3 + n2 + n+ 1≤ 1
n(n+ 1)&242 8!:! n ∈ N∗.
n
n3 + n2 + n+ 1≤ 1
n(n+ 1)⇔ n2(n+ 1) ≤ n3 + n2 + n+ 1⇔ n3 + n2 ≤ n3 + n2 + n+ 1⇔ 0 ≤ n+ 1
01$ 3 -A'.:! &242 8!:! n. B!7!, &$'! C$!4$%2 ()D)D, 2 /34.$∞∑
n=1
n
n3 + n2 + n+ 13 5!6-$47$68$)
!"!## $%&'(%&) *+ ,-./012+%' )3 $%&'(%&) *0 4056)
7 !" #$ %&'&() !"#
∞∑
n=1
un $%# &'()! *#+ ,$! un > 0 -#(# *./. n ! limn→∞
un+1
un
= L.
01*2.
3)4 5 &'()!
∞∑
n=1
un 6.17!(8! &! L < 1;
3))4 5 &'()!
∞∑
n=1
un /)7!(8! &! L > 1;
3)))4 9#/# -./!%.& #:(%#( &! L = 1.
* !"#$%&'()"* E$F2
∞∑
n=1
un 1%2 /34.$ 82' 01$ limn→∞
un+1
un
= L. "68G!, :2:! ε > 0 &!:$%!/
$65!68424 K > 0 82' 01$, &242 8!:! n > K -2'$ 2 :$/.712':2:$
∣
∣
∣
∣
un+1
un
− L
∣
∣
∣
∣
< ε.
E1&!6H2%!/ 01$ L < 1. "68G! $#./8$ q 82' 01$ L < q < 1 $ .//! .%&'.52 01$ q − L < 1.
C!%26:! ε = q − L &!:$%!/ $/54$-$4
∣
∣
∣
∣
un+1
un
− L
∣
∣
∣
∣
< q − L :!6:$ -$%
− (q − L) <un+1
un
− L < q − L !1 − (q − L) + L <un+1
un
< q.
I2 J'8.%2 4$'2KG! 5!65'1L%!/ 01$ un+1 < unq. I$//2 4$'2KG! 8$%!/ 01$
un+1 < unq
un+2 < un+1q < unqq < unq2
un+3 < un+2q < unq2q < unq
3
· · ·un+k < un+(k−1)q < unq
k−1q < unqk
$ 2//.% /15$//.-2%$68$, :$ @!4%2 01$
un+1 + un+2 + un+3 + · · · < unq + unq2 + unq
3 + · · · .
MNO
!"# $%# unq + unq2 + unq
3 + · · · & %'( )&*+# ,#!'&"*+-(. -!' *(/0! |q| < 1 #. 1!*"(2"!.
-!23#*,#2"#4 5))+'. 1#6! 7#!*#'( 84949. ( )&*+#
∞∑
n=1
un -!23#*,# )# L < 1.
:!* !%"*! 6(;!. )%1!2<('!) $%# limn→∞
un+1
un
= L > 1, #2"0! !="#*#'!) un+1 > un 1(*( "!;!
n #. ;#))# '!;!. limn→∞
un 6= 0. >!2)#$%#2"#'#2"#. ( )&*+# 20! 1!))%+ ( -!2;+?0! 2#-#))@*+(
1(*( -!23#*,A2-+(4 B!,!. ( )&*+#
∞∑
n=1
un ;+3#*,# )# L > 1.
5 1(*"# C+++D ;! >*+"&*+! ;# EF56('=#*" ;+/ $%#. )# limn→∞
un+1
un
= 1, #2"0! #)"# -*+"&*+!
& +2-!2-6%)+3!4 G=)#*3# +))! -!2)+;#*(2;! !) #H#'16!)I
∞∑
n=1
1
n2#
∞∑
n=1
1
n. :(*( ('=()
limn→∞
un+1
un
= 1, 1!*&' ( 1*+'#+*( & %'( )&*+# 1. -!' p = 2, -!23#*,#2"# # ( )#,%2;( &
( )&*+# <(*'J2+-( $%# )(=#'!) )#* ;+3#*,#2"#4
!"#$% &'(')* !"#$% % &'()*'(% $+ , -./"01+')2 +!)3$+ " &%#4+'56#&(" $" !*'(+
∞∑
n=1
2n
n.
!"#$%!& 7#'!) $%# un =2n
n# un+1 =
2n+1
n+ 1. B!,!.
un+1
un
=n2n+1
2n (n+ 1)=
n2n2
2n (n+ 1)=
2n
(n+ 1)
# ())+'. 1#6! -*+"&*+! ;# EF56#'=#*". "#'!) $%#
L = limn→∞
un+1
un
= limn→∞
2n
(n+ 1)= 2 > 1.
>!2)#$%#2"#'#2"#. ( )&*+#
∞∑
n=1
2n
n& ;+3#*,#2"#4
!"#$% &'(')+ 7!)3$+ " &%#4+'56#&(" $" !*'(+
∞∑
n=1
1
n!.
!"#$%!& 7#'!) $%# un =1
n!# un+1 =
1
(n+ 1)!# #2"0!
L = limn→∞
un+1
un
= limn→∞
n!
(n+ 1)!= lim
n→∞
1
n+ 1= 0 < 1,
1!*"(2"! ( )&*+#
∞∑
n=1
1
n!-!23#*,#. 1#6( -*+"&*+! ;# EF56#'=#*"4
!"!# $%&'(%&) *+ $,-./0 )- $%&'(%&) *, 1,23
4 !" #$ %&'&() 8+9"
∞∑
n=1
un 30" !*'(+ )"/ :3+ un > 0 ;"'" )%$% n + limn→∞
n√un = L.
7#)<%
KLL
!" # $%&!'
∞∑
n=1
un ()*+'&,' $' L < 1;
!!" # $%&!'
∞∑
n=1
un -!+'&,' $' L > 1;
!!!" ./-/ 0)-'1)$ /2&1/& $' L = 1.
!"#$% &'(')* 3$/*-) ) (&!4%&!) -' 5/6(789 '$46-' / ()*+'&,:*(!/ -/ $%&!'
∞∑
n=1
(
n
2n+ 5
)n
.
!"#$%!& !"#$ %&!
n√un = n
√
(
n2n+5
)n= n
2n+5! '()*+',-# # +.*/0.*# -! 1'&+234 #5/!"#$
%&!
L = limn→∞
n√un = lim
n→∞
n
2n+ 5=
1
2< 1,
! +#,+)&6"#$ %&! ' $0.*!
∞∑
n=1
(
n
2n+ 5
)n
0 +#,7!.8!,/!9
!"#$% &'(')+ ;$46-' / ()*+'&,:*(!/ -/ $%&!'
∞∑
n=1
52n
23n+1.
!"#$%!& !"#$ %&!
n√un =
n
√
52n
23n+1=
52
23+1
n
=25
8.21
n
.
:$$*"4
L = limn→∞
n√un = lim
n→∞
25
8.21
n
=25
8> 1
! ' $0.*!
∞∑
n=1
52n
23n+1-*7!.8!4 (!)# +.*/0.*# -! 1'&+239
!"# $%&'() *( +(&,-) .-)'/'0-) ( 1(23/'0-)
' !"#"$%& '()*() <'=/ un > 0 0/&/ 4)-) n ∈ N∗. >'*)1!*/1)$ ()*+, -".,*/-0- ? $%&!'
-/ @)&1/
∞∑
n=1
(−1)n−1 un = u1 − u2 + u3 − u4 + · · ·+ (−1)n−1 un + · · ·
)6
∞∑
n=1
(−1)n un = −u1 + u2 − u3 + · · ·+ (−1)n un + · · ·
!"#$% &'),'- # $%&!'
∞∑
n=1
(−1)n−1 1
np= 1 − 1
2p+
1
3p− 1
4p+ · · · + (−1)n−1 1
np+ · · · % 61
'A'10B) -' $%&!' /B4'&*/-/C
;<=
!"#!$ %&'()*+,'-./ 0) 12/ 34*.) /56)*'/0/
!"#$%&'#!(# ()*)+ )+ ,-%(.-%)+ *# ,)!/#01!,%2 /%+()+ 2(. ) ')'#!() !3) +3) /4$%*)+ 52-2
+.-%#+ 2$(#-!2*2+6 5)%+ #$#+ #7%0%2' 89# )+ (#-')+ *2 +.-%# ")++#' ()*)+ 5)+%(%/)+: ; +#09%-6
52++2-#')+ 2 /#- 2$09!+ -#+9$(2*)+ 89# +3) /4$%*)+ 52-2 +.-%#+ *# (#-')+ 5)+%(%/)+ # !#02(%/)+:
7 !" #$ %&'(&) *+,-.,/0 1, 2,3453678 !"#$%&'& ()* #+'$& *,-&'"*%*
∞∑
n=1
(−1)n−1 un = u1 − u2 + u3 − u4 + · · ·+ (−1)n−1 un + · · ·
-*, .(&
(i) u1 > u2 > u3 > u4 > · · · (ii) limn→∞
un = 0.
/"-0! #0! 12,$%*# *# #&3($"-&# 4!"4,(#5
7*8 9 #+'$& *,-&'"*%* + 4!"1&'3&"-&:
7;8 9 #!)* <*'4$*, Sn %* #+'$& *,-&'"*%* + -*, .(& 0 < Sn < u1.
9 !"#$%&'()"* <2= >)!+%*#-#')+ 2 +)'2 *)+ 2n 5-%'#%-)+ (#-')+ *2 +.-%# 2$(#-!2*2:
?95)!@2')+ 89# )+ (#-')+ *# )-*#' A'52- *2 +.-%# +3) 5)+%(%/)+ # )+ *# )-*#' 52- +3)
!#02(%/)+: ?#6 5)- 2,2+) ) 5-%'#%-) (#-') ")- !#02(%/)6 %!%,%2-#')+ 2 ,)!(20#' #' u2, 5)%+
2 -#(%-2*2 *# 9' !B'#-) C!%() *# (#-')+ !3) 2"#(2 2 ,)!/#-01!,%2 *2 +.-%#: D#++# ')*)6 )
(#-') u2n−1 . 5)+%(%/) # ) (#-') u2n . !#02(%/): ;++%'6 5#$2 ,)!*%E3) (i) (#')+ 89#
(u1 − u2) > 0, (u3 − u4) > 0, · · · (un − un+1) > 0, · · · (u2n−1 − u2n) > 0
*# ')*) 89#
S2 = u1 − u2 > 0 S4 = S2 + (u3 − u4) > S2 S6 = S4 + (u5 − u6) > S4
# 2++%' +9,#++%/2'#!(#: F)-(2!()6 )G(#')+ 89#
0 < S2 < S4 < .... < S2n.
;%!*26 2++),%2!*) )+ (#-')+ *# )9(-2 ")-'26 )G(#')+ 89#
S2n = (u1 − u2) + (u3 − u4) + ...+ (u2n−1 − u2n)= u1 − (u2 − u3)− (u4 − u5)− ...− (u2n−2 − u2n−1)− u2n
#6 5#$2 ,)!*%E3) (i), ,2*2 (#-') #!(-# 52-1!(#+#+ . 5)+%(%/2: F)-(2!()6 #+(2')+ +9G(-2%!*)
9'2 892!(%*2*# 5)+%(%/2 *# u1, )G(#!*) 9' -#+9$(2*) %!"#-%)- 2 u1, *# ')*) 89# 0 < S2n <
u1.
>)' %++)6 +#09# 89# S2n . $%'%(2*2 # ,)') 0 < S2 < S4 < · · · < S2n, (2'G.' . ')!H()!2:
;++%'6 ,)!,$9A')+ 89# 2 +#891!,%2 *# +)'2+ S2, S4, · · · , S2n ,)!/#-0#6 5#$) I#)-#'2 J:J:K:
?#L2 limn→∞
S2n = S. >)') S2n < u1, +#09# 89# S < u1. ?#!*) S2n+1 = S2n + u2n+1 #
25$%,2!*) 2 ,)!*%E3) (ii), (#')+ 89#
limn→∞
S2n+1 = limn→∞
S2n + limn→∞
u2n+1 = S + 0 = S.
>)!+#89#!(#'#!(# 2+ +)'2+ *# )-*#' A'52- (#' 2 '#+'2 +)'2 *)+ (#-')+ *# )-*#'
52-: M%!2$'#!(#6 ')+(-2-#')+ 89# limn→∞
Sn = S.
>)') limn→∞
S2n = S, *2*) ε > 0 5)*#')+ #!,)!(-2- K1 > 0 (2$ 89# |S2n − S| < ε +#'5-#
89# 2n > K1.
NOP
!"! limn→∞
S2n+1 = S, #$#! ε > 0 %!#&"!' &()!(*+$+ K2 > 0 *$, -.& |S2n − S| < ε
'&"%+& -.& 2n+ 1 > K2.
/!"$(#! K = max {K1, K2} , %$+$ *!#! n > K 0$,& $ #&'12.$,#$#& |Sn − S| < ε. 3!2!4
limn→∞
Sn = S & $ '5+1&∞∑
n=1
(−1)n−1 un 5 )!(0&+2&(*&6
!"#$% &'()'& !"#$% % &'%(')" $' *'+,#+&-. '!&/$' " 0%#1'(23#0+" $" !4(+'
∞∑
n=1
(−1)n−1 n+ 2
n (n+ 1).
!"#$%!& 7$"!' 0&+18)$+ '& un '$*1'9$: *!#$' )!(#1;<&' #! /&!+&"$ =6>?6@6 A *&+"! 2&+$,
#$ '5+1& 5 un =n+ 2
n (n+ 1)> 0 %$+$ *!#! n ∈ N∗. B2!+$4 0$"!' 0&+18)$+ '& un > un+1 %$+$
*!#! n ($*.+$,6 /&"!' -.&
n+ 2
n (n+ 1)>
n+ 3
(n+ 1) (n+ 2)⇔ (n+ 2) (n+ 1) (n+ 2) > n (n+ 1) (n+ 3)⇔ n3 + 5n2 + 8n+ 4 > n3 + 4n2 + 3n⇔ n2 + 5n+ 4 > 0,
-.& 5 0&+#$#&1+! %$+$ *!#! n ($*.+$,6 B''1"4 $ %+1"&1+$ )!(#1;C! #! /&!+&"$ =6>?6@ &'*D
'$*1'9&1*$6 B1(#$4
limn→∞
un = limn→∞
n+ 2
n (n+ 1)= 0.
& &(*C! *!#$' $' &E12F()1$' #! /&!+&"$ =6>?6@ &'*C! '$*1'9&1*$'6 G!#&"!' )!(),.1+ &(*C! -.&
$ '5+1&
∞∑
n=1
(−1)n−1 n+ 2
n (n+ 1)
5 )!(0&+2&(*&6
!"" #$%&' (' )'%*+, (' #&-.&, /0.&,10'%
' !"#"$%& '())() 5'#%)+#")%! !4(+' $' &'()%! $' !+#"+! 6/"+!6/'( 7 &%$" !4(+' 8%()"$"
9%( &'()%! 9%!+&+1%! ' #'2"&+1%!:
B' '5+1&' $,*&+($#$' 'C! )$'!' %$+*1).,$+&' #$' '5+1&' #& *&+"!' #& '1($1' -.$1'-.&+6
!"#$% &'(('* ; !4(+'
∞∑
n=1
sin(nπ6) = 1
2+√32+1+
√32+ 1
2+0− 1
2−√32−1−
√32− 1
2+0+ · · ·
4 /) '<')9=% $' !4(+' $' &'()%! $' !+#"+! 6/"+!6/'(:
7&+&"!' ($ '&-.F()1$ ." *&!+&"$ -.& %&+"1*& 0&+18)$+ '& ."$ '5+1& #& *&+"!' #& '1($1'
-.$1'-.&+ 5 )!(0&+2&(*&6
!" #$ %&''&( >'?"
∞∑
n=1
un /)" !4(+' $' &'()%! $' !+#"+! 6/"+!6/'(: >' " !4(+'
∞∑
n=1
|un|
8%( /)" !4(+' 0%#1'(2'#&' '#&@% " !4(+'
∞∑
n=1
un &"),4) !'(A 0%#1'(2'#&':
>H>
! "#$%#$!& '" % '()*"
∞∑
n=1
|un| +!) ,*-")."#$"& #%,% /!,")"0!' %1)0%) '!2)" % 3!#-").4#3*%
,% '()*" ," '*#%*' 56%*'56")
∞∑
n=1
un.
!"#$% &'((') !"#$ %# &'(")*# +,-.,+ /0( 1 $23!(
∞∑
n=1
(−1)n−1 n+ 2
n (n+ 1)2 4#%5(36(%7(,
8#32"9 1 $23!(
∞∑
n=1
∣
∣
∣
∣
∣
(−1)n−1 n+ 2
n (n+ 1)
∣
∣
∣
∣
∣
=∞∑
n=1
n+ 2
n (n+ 1)%:# 2 4#%5(36(%7(, ; *(!7#3 )#<( 5(3!=413
($$1 1=3"1>:# 0$1%<# # 43!723!# <1 4#")131>:#,
!"#$% &'(('& ?$1%<# # @(#3("1 +,--,A9 ($70<( 1 4#%5(36B%4!1 <1 $23!(
∞∑
n=1
(−1)n−1n3
.
!"#$%!& 7"0!' 56"
∞∑
n=1
∣
∣
∣
(−1)n−1
n3
∣
∣
∣ =∞∑
n=1
1n3 . 8!0! /!,"0!' !2'")-%)& "'$% ( 60% '()*" p 3!0
p = 3 > 1 "& /!)$%#$!& 3!#-")."#$"9 :!.!&∞∑
n=1
(−1)n−1
n3 ( 3!#-")."#$"9 ; 3!#-").4#3*% ,"'$%
'()*" $%02(0 /!," '") "'$6,%,% /"<! $"!)"0% ," :"*2#*$=9
!"#$% &'(('* ?$1%<# # @(#3("1 +,--,A ($70<( 1 4#%5(36B%4!1 <1 $23!(
∞∑
n=1
sin(nx) + 3 cos2(n)
n2.
!"#$%!& 7"0!' 56"
∞∑
n=1
∣
∣
∣
∣
sin(nx) + 3 cos2(n)
n2
∣
∣
∣
∣
=∞∑
n=1
|sin(nx) + 3 cos2(n)|n2
" 3!0! |sin(nx)| ≤ 1 " |cos2(n)| ≤ 1, 6'%#,! /)!/)*",%,"' ," 0>,6<!& '".6" 56"
∣
∣sin(nx) + 3 cos2(n)∣
∣ ≤ |sin(nx)|+∣
∣3 cos2(n)∣
∣ ≤ 1 + 3∣
∣cos2(n)∣
∣ ≤ 1 + 3 = 4,
" "#$?! /!,"0!' 3!#3<6*) 56"
∞∑
n=1
|sin(nx) + 3 cos2(n)|n2
≤∞∑
n=1
4
n2
/%)% $!,! n #%$6)%<9 8!0!∞∑
n=1
4n2 ( 60% '()*" p 3!#-")."#$" @p = 2 > 1), $"0!' 56" % '()*"
∞∑
n=1
∣
∣
∣
∣
sin(nx) + 3 cos2(n)
n2
∣
∣
∣
∣
3!#-")."& /"<! 3)*$()*! ,% 3!0/%)%A?!9
;''*0& % '()*"
∞∑
n=1
sin(nx) + 3 cos2(n)
n2$%02(0 3!#-")."& /"<! 7"!)"0% B9CC9D9
CEF
!"# $%&'() *+),-./*0(1/( 2,13(&4(1/( ( 2,15'2',1*-0(1/(
2,13(&4(1/()
!"#$ %# %#&!'( $)('#$ *+$,-."*/#!"# 0,!1#(2#!"# # 0,!%'0',!*-/#!"# 0,!1#(2#!"#$ 1*/,$
0,!$'%#(*( ,$ #3#/4-,$ *+*'3,5
!"#$% &'()'( !"#$%&'&(!# ) #*'$& +)'(,"$-)
∞∑
n=1
1
n= 1 +
1
2+
1
3+
1
4+ · · ·+ 1
n+ · · ·
./ (!#0')(!# 12& �) #*'$& * %$3&'4&"0&5 6!'*(7 ) #*'$& +)'(,"$-) )80&'")%)7 %)%) 9!'
∞∑
n=1
(−1)n−1 1
n= 1− 1
2+
1
3− 1
4+ · · ·+ (−1)n−1 1
n+ · · ·
* -!"3&'4&"0&7 9&8! 0&!'&() %& :&$;"$0<5 =)(!# (!#0')' 12& ) #*'$&
∞∑
n=1
(−1)n−1 1
n-!"3&'4&
#!; -!"%$>? $#0! *7 9!%&(!# $"0&'@&'$' ") #2) @!'() %& -!"3&'4$'5
!"#$%!& 6*(* /,%'&0*( , 1*-,( %# 0,!1#(27!0'* %#
∞∑
n=1
(−1)n−1 1
n+*$"* (#*2(.4*( ,$ "#(/,$
%#$"* $)('#8 $#4*(*!%, * $,/* %,$ "#(/,$ %# ,(%#/ 9/4*( %* $,/* %,$ "#(/,$ %# ,(%#/ 4*(8
0,!:,(/# $#2.#;
Sn =
(
1 +1
3+
1
5+ · · ·+ 1
2n− 1+ · · ·
)
−(
1
2+
1
4+
1
6+ · · ·+ 1
2n+ · · ·
)
.
<,/, , -#'",( 4,%# ,+$#(1*(8 4,%#/,$ #$0(#1#(
Sn =∞∑
n=1
1
2n− 1−
∞∑
n=1
1
2n
#8 0*%* ./* %#$"*$ $.+=$,/*$ ) %'1#(2#!"#5 >,2,8 "#/,$ ?.# Sn =∞−∞, '$", )8 * $,/* )
'!%#"#(/'!*%*8 $'2!'&0*!%, ?.#8 $# #$0(#1#(/,$
∞∑
n=1
(−1)n−1 1
n
!* :,(/*
∞∑
n=1
(−1)n−1 1
n=
(
1 +1
3+
1
5+ · · ·+ 1
2n− 1+ · · ·
)
−(
1
2+
1
4+
1
6+ · · ·+ 1
2n+ · · ·
)
!*%* 4,%#/,$ *&(/*( $,+(# * $.* 0,!1#(27!0'*5 @$$, ,0,((# 4,(?.# * $)('#
∞∑
n=1
∣
∣
∣
∣
(−1)n−1 1
n
∣
∣
∣
∣
=∞∑
n=1
1
n
!A, 0,!1#(2#5
<,/ +*$# !, #3#/4-, *!"#(',(8 1*/,$ %#&!'( $)('#$ *+$,-."*/#!"# 0,!1#(2#!"# # 0,!%'=
0',!*-/#!"# 0,!1#(2#!"#5
BCD
!"#"$%& '()*(* !"#
∞∑
n=1
un $%# &'()! *! +!(%,& *! &)-#)& .$#)&.$!(/ !-+0,1
2)3 !
∞∑
n=1
|un| 4,-5!(6!/ # &'()! ' *!-,%)-#*# #7&,8$+#%!-+! 4,-5!(6!-+!9
2))3 !
∞∑
n=1
un 4,-5!(6! !
∞∑
n=1
|un| *)5!(6!/ !-+0, # &'()!
∞∑
n=1
un ' *!-,%)-#*# 4,-*)4),-#8:
%!-+! 4,-5!(6!-+!9
!"#$% &'()'* ; &'()!
∞∑
n=1
(−1)n−1 1
n, !&+$*#*# -, <=!%>8, ?9@A9@/ ' 4,-*)4),-#8%!-+!
4,-5!(6!-+! !-.$#-+, .$! # &'()!
∞∑
n=1
sin(nx) + 3 cos2(n)
n2, !&+$*#*# -, <=!%>8, ?9@@9B/ '
#7&,8$+#%!-+! 4,-5!(6!-+!9
!"#$% &'()'+ C8#&&)D.$! # &'()! -$%'()4#
∞∑
n=1
(−1)n−1 n2
n3 + 44,%, #7&,8$+#%!-+! 4,-5!(:
6!-+!/ 4,-*)4),-#8%!-+! 4,-5!(6!-+! ,$ *)5!(6!-+!9
!"#$%&"' !"#$ %&!
∞∑
n=1
∣
∣
∣
∣
(−1)n−1 n2
n3 + 4
∣
∣
∣
∣
=∞∑
n=1
n2
n3 + 4, ! !$'( ) &"( $)*+! ,+-!*.!/'!0 1#+$ (
2&/34# f(x) =x2
x3 + 4) 5#/'6/&( 1(*( '#,# x 6= 3
√−4, !" 1(*'+5&7(* 1(*( '#,# x ≥ 1, )
1#$+'+-( 1(*( '#,# x ≥ 3√−2, !" 1(*'+5&7(* 1(*( x ≥ 1, ! 5#"# f ′(x) =
x(8− x3)
(x3 + 4)2> 0 1(*(
'#,# x > 20 #& $!8(0 7#.# ( 2&/34# f(x) ) ,!5*!$5!/'! 1(*( '#,# x ≥ 2, ! ($$+" 1#,!"#$
(17+5(* # 5*+')*+# ,( +/'!.*(70 ! ,!$'! $!.&! %&!
∫ +∞
2
x2
x3 + 4dx = lim
b→+∞
∫ b
2
x2
x3 + 4dx = lim
b→+∞
1
3ln(x3 + 4)
∣
∣
∣
∣
∣
b
2
= +∞,
#& $!8(0 ( +/'!.*(7 +"1*91*+(0 ! 5#/$!%&!/'!"!/'! ( $)*+!0 ,+-!*.!:
;#*)"0
∞∑
n=1
(−1)n−1 n2
n3 + 4) &"( $)*+! (7'!*/(,( 5#/-!*.!/'!0 1#+$ $('+$2(< ($ 5#/,+3=!$ ,#
'!#*!"( ,! >!+?/+'<0 -+$'# %&!
limn→+∞
n2
n3 + 4= 0 ! un+1 =
(n+ 1)2
(n+ 1)3 + 4≤ n2
n3 + 4= un, 1(*( '#,# n ≥ 2
1#+$ (5+"( -!*+@5("#$ %&! ( 2&/34# f(x) =x2
x3 + 4) ,!5*!$5!/'! 1(*( '#,# x ≥ 2.
;#*'(/'# ( $)*+! ,(,( ) 5#/,+5+#/(7"!/'! 5#/-!*.!/'!:
!"#$% &'()'& C8#&&)D.$! #& &'()!& -$%'()4#& #7#)=, 4,%, #7&,8$+#%!-+! 4,-5!(6!-+!/
4,-*)4),-#8%!-+! 4,-5!(6!-+! ,$ *)5!(6!-+!/ "$&+)D4#-*, &$# (!&>,&+#9
2#3
∞∑
n=2
(−2)n(lnn)n + 2
√n+ 1
273
∞∑
n=1
(−1)n24√n3 + 2n
!"#$%&"' A(B C/(7+$(/,# ( 5#/-!*.D/5+( (?$#7&'( '!"#$
∣
∣
∣
∣
(−2)n(lnn)n + 2
√n+ 1
∣
∣
∣
∣
=2n
(lnn)n + 2√n+ 1
≤ 2n
(lnn)n
!"
!"#$%&'( ( )*+)* '% ,%#-. )*/(+
L = limn→∞
n
√
2n
(lnn)n= lim
n→∞
2
lnn= 0.
0(/( L < 1 % +1,#*∞∑
n=2
2n
(lnn)n$(&2*,3*4 5(3(. !*"( )*+)* '% $(/!%,%67(. % +1,#* '%'%
$(&2*,3* %8+("9)%/*&)*4
:8; &%"#+%&'( % $(&2*,3<&$#% %8+("9)% )*/(+
∣
∣
∣
∣
(−1)n24√n3 + 2n
∣
∣
∣
∣
=2
4√n3 + 2n
≤ 24√n3
,
$(/ #++( &%'% !('*/(+ $(&$"9#,. !(#+ % +1,#* '%'% 1 /*&(, =9* 9/% +1,#* ! '#2*,3*&)*4
>(,1/. (8+*,2* =9*
24√n3 + 2n
=2
[n3(1 + 2n2 )]
1
4
=2
n3
4 (1 + 2n2 )
1
4
* 1 ≤ (1 +2
n2)1
4 ≤ 31
4 . 5(3(.
24√n3 + 2n
≥ 24√3n
3
4
,
*. !(, $(/!%,%67(. % +1,#* '%'% &7( $(&2*,3* %8+("9)%/*&)*4
&%"#+%&'( % $(&2*,3<&$#% $(&'#$#(&%". 9+%&'( ( ?*(,*/% '* 5*#8&#)-. !(#+ % +1,#* '%'%
1 %")*,&%'%. )*/(+ limn→∞
24√n3 + 2n
= 0 * an =2
4√n3 + 2n
1 '*$,*+$*&)*4
>(,)%&)(. % +1,#* '%'% 1 $(&'#$#(&%"/*&)* $(&2*,3*&)*4
!"# $%&'() *( +,-./()
0(&+#'*,%&'( %+ @9&6A*+ fi : R → R '*B&#'%+ !(, f0 (x) = 1, f1 (x) = x, f2 (x) = x2,
f3 (x) = x3, f4 (x) = x4, · · · , fn (x) = xn, · · · , !('*/(+ *+$,*2*, % +(/%
S (x) = f0 (x) + f1 (x) + f2 (x) + f3 (x) + f4 (x) + · · ·+ fn (x) + · · ·= 1 + x+ x2 + x3 + x4 + · · ·+ xn + · · ·
C++% +(/% #&B&#)% 1 9/ *D*/!"( '* +1,#* '* @9&6A*+. !(#+ ( +*9 )*,/( 3*,%" '*!*&'* '*
9/% 2%,#E2*" ,*%" x. F%#+ 3*,%"/*&)*. '*B&#/(+ +1,#* '* @9&6A*+ $(/( +*39*4
!"#"$%& '()*() !"#$%"&$#' '()%! *! +,"-.!' & /#*& '()%! "& 0,&1 # /!)$# 2!)&1 ( ,$&
+,"-3# *& 4&)%54!1 )!&1 x ! & *!"#/&)!$#' 6#)
∞∑
n=0
un (x) = u0 (x) + u1 (x) + u2 (x) + · · ·+ un (x) + · · ·
!"#!$ %&'()*+,'-./ 0) 12*.)1 0) 34'56)1
0(/( &( *+)9'( '%+ +1,#*+ &9/1,#$%+. *+)%/(+ #&)*,*++%'(+ &% $(&2*,3<&$#% '%+ +1,#*+ '*
@9&6A*+4 G/% +1,#* '* @9&6A*+. +* @(, $(&2*,3*&)*. $(&2*,3#,E !%,% 9/% @9&67(4 #/%3*/
HIJ
! "# # $#%&' ! x ()*# +,'-! ! .)(/0!+ , )*# +,'-! ()*,'-"# 1)! 2& ! +!' "&($!'3!(4! &)
-$!'3!(4!5 6&' !7!*2%&8 2#'# "# # $#%&' ! x, # +,'-!
∞∑
n=0
xn = 1 + x+ x2 + x3 + x4 + · · ·+ xn + · · ·
, )*# +,'-! 3!&*,4'-"# !8 2&'4#(4&8 "&($!'3! +! |x| < 1 ! -$!'3! "#+& "&(4'9'-&5 :9 +)# +&*#
+!'9 # .)(/;& S (x) =1
1− x, +! |x| < 1. <++& +-3(-="# 1)! )*# +,'-! ! .)(/0!+ "&($!'3!(4!8
"&($!'3! 2#'# )* !4!'*-(# & "&(>)(4& ! $#%&'!+ ! x, !(&*-(# & &*?(-& &) -(4!'$#%&
! "&($!'3@("-#5
!"#"$%& '()*(* !"#
∞∑
n=0
un (x) $%# &'()! *! +$,-.!&/ 0!,1%),#%1& *1%2,)1 1$ ),3!(4
5#61 *! 71,5!(89,7)# *# &'()! #1 71,"$,31 *! 31*1& 1& 5#61(!& *! x :#(# 1& ;$#)& # &'()! '
71,5!(8!,3! ! *!,1%),#%1& (#)1 *! 71,5!(89,7)# < *)&3=,7)# !,3(! 1 7!,3(1 ! #& !>3(!%)*#*!&
*1 ),3!(5#61 71,5!(89,7)#/
!"#$% &'()'* ? (#)1 *! 71,5!(89,7)# *# &'()!
∞∑
n=0
xn' R = 1 ! 1 &!$ ),3!(5#61 *! 71,4
5!(89,7)# ' I = (−1, 1) . @#(# 31*1 x ∈ (−1, 1) 3!%4&! ;$!∞∑
n=0
xn =1
1− x.
!"#$% &'()'& 0!3!(%),! 1 ),3!(5#61 ! 1 (#)1 *! 71,5!(89,7)# *# &'()!
∞∑
n=1
cos(x) + sin(x)
n4 + n.
!"#$%&"' A(#%-+#( & # "&($!'3@("-# #B+&%)4# # +,'-!8 4!*&+ 1)!
∣
∣
∣
∣
cos(x) + sin(x)
n4 + n
∣
∣
∣
∣
=|cos(x) + sin(x)|
n4 + n≤ |cos(x)|+ |sin(x)|
n4 + n≤ 2
n4 + n≤ 2
n4
! "&*&
∞∑
n=1
2
n4, )*# pC+,'-! "&($!'3!(4!8 "&("%)?*&+8 2&' "&*2#'#/;&8 1)! # +,'-! # # ,
#B+&%)4#*!(4! "&($!'3!(4!5 D) +!>#8 # +,'-!
∞∑
n=1
cos(x) + sin(x)
n4 + n"&($!'3! 2#'# 4& & $#%&'
'!#% ! x. A++-*8 & -(4!'$#%& ! "&($!'3@("-# !+4# +,'-! , R ! +!) '#-& ! "&($!'3@("-# ,
-(=(-4&5
!"# $%&'() *( +,-./0'1)
A+ +,'-!+ ! 2&4@("-#+ +;& #+ +,'-!+ ! .)(/0!+ 1)! #2#'!"!* "&* *#-+ .'!1)@("-# (&+
2'&B%!*#+ ! *#4!*94-"# ! !(3!(E#'-#8 2&-+ +;& F4!-+ (# -(4!3'#/;& ! .)(/0!+ 1)! (;&
2&++)!* #(4- !'-$# #+ !%!*!(4#'!+8 (# '!+&%)/;& ! !1)#/0!+ -.!'!("-#-+ ! 4#*B,* 2#'#
#2'&7-*#' .)(/0!+ 2&' 2&%-(G*-&+ H"-!(4-+4#+ .#I!* -++& 2#'# +-*2%-="#' !72'!+0!+ "&*2%!7#+8
2'&3'#*# &'!+ .#I!* -++& 2#'# '!2'!+!(4#' .)(/0!+ !* "#%")%# &'#+ ! "&*2)4# &'!+J5 K*
$-+4# -++&8 $#*&+ #' #4!(/;& !+2!"-#% #& !+4) & #+ L,'-!+ ! 6&4@("-#+5
!"#"$%& '()+() A%# &'()! *! :139,7)#& ' $%# &'()! 7$"1& 3!(%1& !,5165!% #:!,#&
:139,7)#& *! x %$63):6)7#*#& :1( 71!B7)!,3!& 71,&3#,3!& cn, 1$ &!"#C $%# &'()! *! :139,7)#&
' !&7()3# ,# +1(%#
∞∑
n=0
cnxn = c0 + c1x+ c2x
2 + c3x3 + · · ·+ cnx
n + · · · .
MNO
!"#$% &'()'* !"#$%
∞∑
n=0
xn&' ()%*+,' -./0.1 " 2*3 !"#$% &% +'4567$3! '6&% 4'&'! '!
7'%87$%64%! cn !9' $:23$! 3 1. ;< 3 !"#$%∞∑
n=1
cos(x) + sin(x)
n4 + n&' ()%*+,' -./0.- 69' " 2*3
!"#$% &% +'4567$3!= +'$! !%2! 4%#*'! 69' %6>',>%* 3+%63! +'4567$3! &% x.
!+,!-./01% &'()'2 ?3#3 @2% '! #%!2,43&'! 364%#$'#%! +'!!3* !%# 2!3&'! !%* *2&36A3! 63!
6'43AB%!= >3*'! 3&*$4$# @2% un(x) = cnxn+3#3 ' 73!' &3! !"#$%! &% +'4567$3!.
!"#!# $%&'())& *+%+ ,(-(%./0+% & /0-(%1+2& ( & %+/& ,( '&01(%3405
'/+ ,( 6.+ )7%/( ,( *&-40'/+)
!"#"$%&'() *( +,"!-,"*( .) / 01#%&2),! *3 .) 4%3+56 7%,% % +*89),:;8+"% %2(*#3!%<
!*&%8.* limn→∞
∣
∣
∣
∣
un+1
un
∣
∣
∣
∣
*3 limn→∞
(
n
√
|un|)
*8.) un = cnxn. 4%(* * #"&"!) )="(!% 9%#) %
+*8.">?* .*( +,"!-,"* 3(%.*@ A& B3%#B3), +%(* !),)&*( B3)
limn→∞
∣
∣
∣
∣
un+1
un
∣
∣
∣
∣
= limn→∞
∣
∣
∣
∣
cn+1xn+1
cnxn
∣
∣
∣
∣
= |x|L
*8.)
L = limn→∞
∣
∣
∣
∣
cn+1
cn
∣
∣
∣
∣
.
/)(() &*.*< * ,%"* ) * "8!),9%#* .) +*89),:;8+"% (),?* *2!".*( ,)(*#9)8.* % "8)B3%>?*
|x|L < 1, B3) 8*( .C |x| < 1L, *3 ()D%< * ,%"* .) +*89),:;8+"% -
R =1
L.
!+,!-./01% &'()'& C'*' ' 7#$4"#$' &% D E ,3*F%#4 " $67'67,2!$>' @236&' ' ,$*$4% &3 #3G9'
" $:23, 3 /= 63&3 +'&%*'! 38#*3# !% |x|L = 1. !!$*= &%>%*'! >%#$873# !% 3 !"#$% 7'6H
>%#:% +3#3 x =1
L% x = − 1
L. I%$43 %!43 >%#$873A9'= +'&%H!% %!43F%,%7%# ' $64%#>3,' &%
7'6>%#:567$3.
!"#$% &'()'3 D%4%#*$6% ' $64%#>3,' % ' #3$' &% 7'6>%#:567$3 &3 !"#$%
∞∑
n=0
3nxn
5n (1 + n2).
!"#$%!& 17#"+%8.* * +,"!-,"* .) /01#%&2),! 7%,% % +*89),:;8+"% %2(*#3!%< !)&*( B3)
limn→∞
∣
∣
∣
∣
un+1
un
∣
∣
∣
∣
= limn→∞
∣
∣
∣
∣
∣
∣
∣
∣
∣
3n+1xn+1
5n+1(
1 + (n+ 1)2)
3nxn
5n (1 + n2)
∣
∣
∣
∣
∣
∣
∣
∣
∣
= limn→∞
∣
∣
∣
∣
5n3n3xnx (1 + n2)
5n5 (n2 + 2n+ 2) 3xn
∣
∣
∣
∣
= limn→∞
∣
∣
∣
∣
3x (1 + n2)
5 (n2 + 2n+ 2)
∣
∣
∣
∣
= |x| limn→∞
∣
∣
∣
∣
3 (1 + n2)
5 (n2 + 2n+ 2)
∣
∣
∣
∣
=3
5|x|
1(("&< % (-,") +*89),:",C ()
3
5|x| < 1, *3 ()D%< () |x| < 5
3. E*,!%8!*< * ,%"* .) +*89),:;8+"%
- R = 53.
F% ()B3;8+"% .)9)&*( 9),"G+%, () % (-,") +*89),:) 7%,% x = −5
3) x =
5
3.
HIJ
• ! x = −5
3, "!#$% & %'()!
∞∑
n=0
3n(
−53
)n
5n (1 + n2)=
∞∑
n=0
(−1)n 3n5n
5n (1 + n2) 3n=
∞∑
n=0
(−1)n 1
(1 + n2).
*+! ,$-.!(/!0 1!2$ ,()"'()$ 3! 4!)5-)"67
• ! x =5
3"!#$% & %'()!
∞∑
n=0
3n(
53
)n
5n (1 + n2)=
∞∑
n=0
3n5n
5n (1 + n2) 3n=
∞∑
n=0
1
(1 + n2).
*+! ,$-.!(/! 1$( ,$#1&(&89$0 1$)%
∞∑
n=0
1
(1 + n2)≤ 1 +
∞∑
n=1
1
n2.
:$-,2+%9$; < (&)$ 3! ,$-.!(/=-,)& 3& %'()!
∞∑
n=0
3nxn
5n (1 + n2)' R =
5
3! $ %!+ )-"!(.&2$
3! ,$-.!(/=-,)& ' −5
3≤ x ≤ 5
3.
!"#$% &'()'* !"!#$%&'# ( %&"!#)'*( ! ( #'%( +! ,(&)!#-.&,%' +' /0#%!
∞∑
n=0
n!xn.
!"#$%!& >12),&-3$ -$.&#!-"! $ ,()"'()$ 3! ? @>2!("0 "!#$% *+!
limn→∞
∣
∣
∣
∣
un+1
un
∣
∣
∣
∣
= limn→∞
∣
∣
∣
∣
(n+ 1)!xn+1
n!xn
∣
∣
∣
∣
= limn→∞
(n+ 1) |x| ={
0, %! x = 0∞, %! x 6= 0
.
>%%)#0 & %'()! 3&3& ,$-.!(/! &1!-&% *+&-3$ x = 0. A$("&-"$0 $ %!+ )-"!(.&2$ 3! ,$-B
.!(/=-,)& ' I = {0} ! R = 0 ' $ %!+ (&)$ 3! ,$-.!(/=-,)&7
!"#!$ %&'() *) +,-./0(12 0)/-'1*1 )3 x = a
' !"#"$%& '()*(+ !&($%&'$(/ /0#%! +! 1(".&,%'/ ,!&"#'+' !$ x = a 2 "(+' /0#%! +'
3(#$'
∞∑
n=0
cn (x− a)n .
A&(& $5"!( $ (&)$ ! $ )-"!(.&2$ 3! ,$-.!(/=-,)& 3&% %'()!% !# (x− a) , 5&%"& C&6!( z =
(x− a) ! !-,$-"(&( $ )-"!(.&2$ 3! ,$-.!(/=-,)& 1&(& & %'()!
∞∑
n=0
cnzn. >1D% !%"& !"&1&0
%+5%")"+)B%! z 1$( (x− a) -& )-!*+&89$ −R < z < R.
!"#$% &'()'(+ !"!#$%&'# ( #'%( ! ( %&"!#)'*( +! ,(&)!#-.&,%' +' /0#%!
∞∑
n=0
2 (x− 5)
n2 + 3
n
.
!"#$%!& !E& z = (x− 5)7 F-"9$ 1$3!#$% !%,(!.!(
∞∑
n=0
2 (x− 5)
n2 + 3
n
=∞∑
n=0
2zn
n2 + 3.
G%&-3$ $ "!$(!#& 3! ?@>2!(" "!#$% *+!
HIJ
limn→∞
∣
∣
∣
∣
un+1
un
∣
∣
∣
∣
= limn→∞
∣
∣
∣
∣
∣
∣
∣
∣
∣
2zn+1
(n+ 1)2 + 32zn
n2 + 3
∣
∣
∣
∣
∣
∣
∣
∣
∣
= limn→∞
∣
∣
∣
∣
∣
(n2 + 3) 2zn+1
(
(n+ 1)2 + 3)
2zn
∣
∣
∣
∣
∣
= limn→∞
(n2 + 3) |z|(n2 + 2n+ 4)
= |z| limn→∞
n2 + 3
n2 + 2n+ 4= |z|
!""#$ ! "%&# '()* &+ " |z| < 1. ,(&-!)-(. ( " / &!#( 0 '()* &+1)'#! % R = 1. 2!
" 3/1)'#!. 0 * $(" * '!& " ! "%&# '()* &+ 5!&! z = −1 z = 1.
• 6 z = −1 - $(" ! "%&#
∞∑
n=0
2zn
n2 + 3=
∞∑
n=0
2 (−1)nn2 + 3
=∞∑
n=0
(−1)n 2
(n2 + 3)
3/ '()* &+ . 5 7( - (& $! 0 8 #9)#-:;
• 6 z = 1 - $(" ! "%&#
∞∑
n=0
2zn
n2 + 3=
∞∑
n=0
2(1)n
n2 + 3=
∞∑
n=0
2
(n2 + 3).
3/ '()* &+ 5(& '($5!&!<=( '($ /$! p−"%&# . 5(#"
∞∑
n=0
2
(n2 + 3)≤ 2
3+
∞∑
n=1
2
n2.
>()'7/"=(? @ &!#( 0 '()* &+1)'#! 0! "%&#
∞∑
n=0
2zn
n2 + 3% R = 1 ( " / #)- &*!7( 0
'()* &+1)'#! % −1 ≤ z ≤ 1. 6/9"-#-/#)0( z 5(& x− 5, (9- $("
4 ≤ x ≤ 6,
3/ % ( #)- &*!7( 0 '()* &+1)'#! 0! "%&#
∞∑
n=0
2 (x− 5)
n2 + 3
n
.
!"#!"" $%&'(&)(*+*, *+ -%.+ *, ).+ /01(, *, 2)&34,-!
6!9 $(" 0( >A7'/7( B 3/ ! "($! 0 /$ )C$ &( 4)#-( 0 D/)<E " '()-F)/!" % '()-F)/!;
,(&%$. " ! "($! )*(7* & #)4)#-(" - &$(". " / & "/7-!0( 5(0 )=( " & '()-F)/(; G H!$("
/$ I $57( ()0 #""( ('(&& ;
!"#$% &'()'(* !"#$% &'% ( ")$*%
∞∑
n=1
(
x1
2n+1 − x1
2n−1
)
+!,-%$.% /($( '0( 1',23! 4%"5
+!,#6,'(7
!"#$%!& J"'& * )0( ! "($! 0(" n−5&#$ #&(" - &$(" 0 "-! "%&#
Sn (x) =(
x1
3 − x)
+(
x1
5 − x1
3
)
+(
x1
7 − x1
5
)
+ · · ·+(
x1
2n+1 − x1
2n−1
)
7#$#)!)0( (" 5!&1)- " ". (9- $(" 3/ Sn (x) = −x+ x1
2n+1 . K""#$.
BLL
S(x) = limn→∞
Sn (x) = limn→∞
(
−x+ x1
2n+1
)
=
{
1− x, ! x 6= 00, ! x = 0.
"#$%&'%#( limn→∞
Sn (x) !)* %! +&$& %#,# x ∈ R ! & -$*! ,! ./'01! ,&,& - 2#'3!$4!'%!5
6#%! 7/! & #8& ,! %& -$*! - /8& ./'09# ,! 2#'%:'/& !8 x = 0, !'7/&'%# 7/! 2&,& /8
,! !/ %!$8# !$& 2#'%:'/#5 ;< !$3! &*',& 7/! & -$*! !8 7/! %9# !" # $%& '#()* +*
,"-. /)&'5
!"#!"$ %&'()*+,- .& /0* 12'(& .& 3/4+5&1 6-4784/*1
6# =>?2/?# @( 3*8# 7/! & ,!$*3&,& ,! /8& #8& A'*%& ,! ./'01! - *4/&? B #8& ,&
,!$*3&,& 5 6# !'%&'%#( ! %*3!$8# /8& 7/&'%*,&,! *'A'*%& ,! ./'01! ( ! & +$#+$*!,&,!
+#,! ,!*)&$ ,! !$ 3>?*,&5 C& 8! 8& .#$8&( & ,!$*3&,& ,! /8& -$*! ,! ./'01! 2#'3!$4!'%!
+#,! !$ ,*3!$4!'%!5 D!E&8# /8 !)!8+?#F
!"#$% &'()'() !"#$%&'& ( #)'$&
∞∑
n=1
sin(n4x)
n2. *!#+'& ,-& &#+( ) -.( #)'$& /!"0&'1&"+& &
,-& ( #)'$& %& #-(# %&'$0(%(# ) %$0&'1&"+&2
0"1$2!"3 =#8# |sin(n4x)| ≤ 1 +&$& %#,# n '&%/$&? ! %#,# x $!&?( !4/! 7/!
∣
∣
∣
∣
sin(n4x)
n2
∣
∣
∣
∣
=|sin(n4x)|
n2≤ 1
n2
! +#$ 2#8+&$&09# 2#8 /8& pG -$*! 2#'3!$4!'%! Hp = 2I( +#,!8# 2#'2?/*$ 7/! & -$*! ,&,& -
&< #?/%&8!'%! 2#'3!$4!'%!5 J*',&( ! %& -$*! 2#'3!$4! +&$& %#,# 3&?#$ $!&? ,! x. K!E& S(x)& #8& ,! %& -$*!( #/ !E&(
S(x) =∞∑
n=1
sin(n4x)
n2=
sin x
12+
sin(24x)
22+
sin(34x)
32+
sin(44x)
42+ · · ·+ sin(n4x)
n2+ · · ·
,!$*3&',# %!$8# & %!$8# ! %& #8&( %!8# 7/!
S ′ (x) =cosx
12+
24 cos(24x)
22+
34 cos(34x)
32+
44 cos(44x)
42+ · · ·+ n4 cos(n4x)
n2+ · · ·
= cos x+ 22 cos(24x) + 32 cos(34x) + 42 cos(44x) + · · ·+ n2 cos(n4x) + · · ·
! &+?*2&',# !8 x = 0, #<%!8#
S ′ (0) = cos 0 + 22 cos 0 + 32 cos 0 + 42 cos 0 + · · ·+ n2 cos 0 + · · ·= 12 + 22 + 32 + 42 + · · ·+ n2 + · · ·
7/! - /8& !7/L'2*& ,! #8& ,*3!$4!'%!5 J *8( & -$*! ,! ./'01! 2#'3!$4! +&$& x = 0,!'7/&'%# 7/! & ,!$*3&,& ,! %& -$*! ,*3!$4! !8 x = 0. ;< !$3! 7/! & -$*! !8 7/! %9# !"
# $%& '#()* +* ,"-. /)&'5
C& 8! 8& .#$8& 7/! '& ,!$*3&,&( & *'%!4$&09# ,! /8& -$*! ,! ./'01! %&8<-8 !)*4!
2/*,&,# 5 M'7/&'%# 7/! & *'%!4$&? ,! /8& #8& A'*%& ,! ./'01! - *4/&? & #8& ,& *'%!4$&* (
# 8! 8# +#,! '9# !$ 3>?*,# +&$& /8& 7/&'%*,&,! *'A'*%& ,! ./'01! 5
6# !'%&'%# * %# '9# #2#$$!$> 7/&',# ! %$&%&$ ,! -$*! ,! +#%L'2*& ( #/ !E&( 7/&',#
/8& -$*! ,! +#%L'2*& .#$ 2#'3!$4!'%! +#,!G ! !.!%/&$ & ,!$*3&09# ! & *'%!4$&09# %!$8# &
%!$8# 7/! & '#3& -$*! #<%*,& +#$ ! %! +$#2! # %&8<-8 !$9# 2#'3!$4!'%! ( 2#8 #
8! 8# $&*# ,! 2#'3!4L'2*&( 2#'.#$8! 3!$!8# & !4/*$5
NOO
!" #$%&'&()$*+,- & .(/&0'*+,- 1& 23'$&4 1& 5-/6()$*4
!"#$ %& '#$ !()*& %& +",-./*$! ( '#$ 0'.12" f(x) =∞∑
n=0
cn (x− a)n , /'3" %"#4.*" (
" *.,&)5$6" %& /".5&)7-./*$ %$ !()*&8 9&.,)" %&!,& *.,&)5$6": $ %&)*5$12" & $ *.,&7)$12" %& f
"/"))& ,&)#" $ ,&)#": "' !&3$: +"%&;!& %&)*5$) & *.,&7)$) /$%$ ,&)#" *.%*5*%'$6 %$ !()*&: %&
$/")%" /"# " )&!'6,$%" $<$*="8
!" #$ %&'%&' !"#
∞∑
n=0
cn (x− a)n $%# &'()! *! +,-./0)#& 0,% (#), *! 0,/1!(2./0)#
R > 0. 3/-4, # 5$/64, f *!7/)*# +,(
f(x) = c0 + c1(x− a) + c2(x− a)2 + · · · =∞∑
n=0
cn (x− a)n
' *)5!(!/0)81!9 :! +,(-#/-, 0,/-;/$#< /, )/-!(1#9, (a−R, a+R) !
!" f ′(x) = c1 + 2c2(x− a) + 3c3(x− a)2 + · · · =∞∑
n=1
ncn (x− a)n−1
!!" f”(x) = 2c2 + 6c3(x− a) + · · · =∞∑
n=2
n(n− 1)cn (x− a)n−2
! #&&)% +,( *)#/-!= >9'% *)&&,? -,%#/*, C = K + ac0, -!%@&! A$!
!!!"
∫
f(x)dx = C + c0(x− a) + c1(x− a)2
2+ c2
(x− a)3
3+ · · · = C +
∞∑
n=0
cn(x− a)n+1
n+ 1
B& (#),& *! 0,/1!(2./0)# *#& &'()!& *#& !A$#6C!& :)<? :))< ! :)))< &4, -,*,& )2$#)& # R.
! !"#$%&'( )*+)*, 3%D,(# , -!,(!%# #/-!(),( *)2# A$! , (#), *! 0,/1!(2./0)# +!(%#/!0!
, %!&%, A$#/*, $%# &'()! *! +,-./0)#& ' *)5!(!/0)#*# ,$ )/-!2(#*#? !##$ %&$ #!'%!()*
+,- $ !%.-/0*1$ 2- )$%0-/'3%)!* 4-/5*%-6* $ 5-#5$= E,*! ,0,((!( *! # &'()! )/)0)#9
0,/1!(2)( !% $% !F-(!%, !/A$#/-, A$! # &'()! *)5!(!/0)#*# *)1!(2! /!&&! +,/-,=
-"./0( )*+)*1 3F+(!&&!
1
(1− x)20,%, $%# &'()! *! +,-./0)#& ! *!-!(%)/! &!$ (#), *!
0,/1!(2./0)#=
7$1,6&$8 >" ?=&#+6" @8AB8C 5*#"! D'&: !& x ∈ (−1, 1) &.,2"
1
1− x= 1 + x+ x2 + x3 + · · · =
∞∑
n=0
xn.
9*0&)&./*$.%" /$%$ 6$%" %&!!$ &D'$12": "<,&#"! D'&
1
(1− x)2= 1 + 2x+ 3x2 + 4x3 + · · · =
∞∑
n=1
nxn−1.
E"%&#"! %&!6"/$) " 4.%*/& %" /".,$%") ,)"/$.%" n +") n+1, &!/)&5&.%" $ )&!+"!,$ /"#"
1
(1− x)2=
∞∑
n=0
(n+ 1)xn.
9& $/")%" /"# " F&")&#$ @8A@8A: " )$*" %& /".5&)7-./*$ %$ !()*& %*0&)&./*$%$ ( " #&!#"
D'& " )$*" %& /".5&)7-./*$ %$ !()*& ")*7*.$6: $ !$<&): R = 1. G 6&*,") +"%&)H 5&)*I/$) D'& "
*.,&)5$6" %& /".5&)7-./*$ %$ !()*& "<,*%$ ( $<&)," ."! &=,)&#"!: "' !&3$: ( " *.,&)5$6" (−1, 1).
JKA
!"#$% &'(&') !"#$%%$
x5
(1− 3x)2&'(' )(* %+#,$ -$ "'./0&,*% $ -$.$#(,0$ %$) ,0.$#1*2'
-$ &'01$#3/0&,*4
!"#$%!& ! "#$%&'! ()*()+ ,-%!. /0$1 &232 x ∈ (−1, 1) 4 ,5'-6! /0$
1
(1− x)2=
∞∑
n=0
(n+ 1)xn.
73!8296! x &!3 3x $% 2%:!. !. '26!. 6$..2 -;02'626$1 !:<$%!.
1
(1− 3x)2=
∞∑
n=0
(n+ 1)(3x)n =∞∑
n=0
3n(n+ 1)xn
$ $..2 .43-$ 8!9,$3;$ .$ 3x ∈ (−1, 1), !0 .$=21 .$ x ∈ (−13, 13). >;!321 &232 !:<$3 2 .43-$
6$.$=262 :2.<2 %0'<-&'-823 2 .43-$ 28-%2 &!3 x5, !:<$96!
x5
(1− 3x)2= x5
∞∑
n=0
3n(n+ 1)xn =∞∑
n=0
3n(n+ 1)xn+5.
?0<32 @!3%2 6$ $.83$,$3 $.<2 .43-$ 4
x5
(1− 3x)2=
∞∑
n=5
3n−5(n− 4)xn
$ .$0 -9<$3,2'! 6$ 8!9,$3;A98-2 4 (−13, 13).
!"#$% &'(&'& 0&'0.#$ * #$"#$%$0.*56' $( %+#,$% -$ "'./0&,*% "*#* f(x) = ln(1− x).
!"#$%!& !<$%!. -9-8-2'%$9<$ /0$1 &$'! "#$%&'! ()*()+ !:<$%!. /0$
f ′(x) =−1
1− x=
∞∑
n=0
−xn
$ -9<$;3296! 2%:!. !. '26!. 6$..2 $/02BC!1 8!% ! 20#D'-! 6! 7$!3$%2 ()*()*1 !:<$%!. /0$
f(x) =
∫ −11− x
dx = C +∞∑
n=0
−xn+1
n+ 1= C −
∞∑
n=1
xn
n.
E232 6$<$3%-923 ! ,2'!3 6$ C, 8!'!82%!. x = 0 9$..2 $/02BC! $ $98!9<32%!. C − 0 =f(0) = ln 1 = 0. >..-%
ln(1− x) = −∞∑
n=1
xn
n= −x− x2
2− x3
3− · · · .
? 32-! 6$ 8!9,$3;A98-2 6$..2 .43-$ 4 ! %$.%! /0$ ! 62 .43-$ !3-;-92'1 R = 1, &!34% !
-9<$3,2'! 6$ 8!9,$3;A98-2 4 I = [−1, 1). F$3-G/0$H !<$ ! /0$ 28!9<$8$ /0296! 8!'!82%!. x = 1
29! 3$.0'<26! 6! "#$%&'! ()*()() I!%!
ln 12= − ln 2, ,$%!. /0$
ln 2 =1
2+
1
8+
1
24+
1
64+ · · · =
∞∑
n=1
1
n2n.
?0 .$=21 0.296! $.<2 .43-$ 6$ @09BJ$. !:<-,$%!. 2 .!%2 62 .43-$ 90%43-82
∞∑
n=1
1
n2n.
KLK
!"# $%&'() *( +,-./&
!"#$%&'& ()* +(",-! f (x) & #&.* a () '&*/ 0(*/0(&'1 2'&3&"%&4#& &"5!"3'*' ()* #6'$&
%& 7!38"5$*# %* +!')*
∞∑
n=0
cn (x− a)n 0(& 5!"9$'.* 7*'* f, !( #&.*: 3*/ 0(&
f (x) =∞∑
n=0
cn (x− a)n .
;) !(3'*# 7*/*9'*#: 0(&'&)!# 0(&
f (x) = c0 + c1 (x− a) + c2 (x− a)2 + c3 (x− a)3 + · · ·+ cn (x− a)n + · · · <=1>?1>@
A##$): 7'&5$#*)!# %&3&')$"*' !# 5!&B5$&"3&# c0, c1, c2, · · ·
• 2'$)&$'! %&3&')$"*)!# c0, 3!)*"%! x = a &) <=1>?1>@1 CD3&)!#
f (a) = c0 + c1 (a− a) + c2 (a− a)2 + c3 (a− a)3 + · · ·+ cn (x− a)n + · · ·
%!"%& 9&)
f (a) = c0.
• E&3&')$"*)!# * %&'$9*%* %* +(",-! <=1>?1>@ & "* #&0(8"5$* *7/$5*)!# &) x = a 7*'*
!D3&' c1, !( #&.*:
f ′ (x) = c1 + 2c2 (x− a) + 3c3 (x− a)2 + · · ·+ ncn (x− a)n−1 + · · ·
f ′ (a) = c1 + 2c2 (a− a) + 3c3 (a− a)2 + · · ·+ ncn (a− a)n−1 + · · ·%!"%& 9&)
f ′ (a) = c1.
• E&3&')$"*)!# * #&F("%* %&'$9*%* %* +(",-! <=1>?1>@ & "* #&0(8"5$* *7/$5*)!# &)x = a 7*'* !D3&' c2, $#3! 6:
f ′′ (x) = 2c2 + 3 · 2c3 (x− a) + 4 · 3c4 (x− a)2 + · · ·+ n(n− 1)cn (x− a)n−2 + · · ·
f ′′ (a) = 2c2 + 3 · 2c3 (a− a) + 4 · 3c4 (a− a)2 + · · ·+ n(n− 1)cn (a− a)n−2 + · · ·%!"%& 9&)
f ′′ (a) = 2c2 !( c2 =f ′′ (a)
2!.
• E&3&')$"*)!# * 3&'5&$'* %&'$9*%* %* +(",-! <=1>?1>@ &: "* #&0(8"5$* f (3) (a) 7*'* !D3&'c3. G&)!#
f (3) (x) = 3·2c3+4·3·2c4 (x− a)+5·4·3c5 (x− a)2+· · ·+n(n−1)(n−2)cn (x− a)n−3+· · ·
f (3) (a) = 3·2c3+4·3·2c4 (a− a)+5·4·3c5 (a− a)2+· · ·+n(n−1)(n−2)cn (a− a)n−3+· · ·%!"%& 9&)
f (3) (a) = 3 · 2c3 !( c3 =f (3) (a)
3!.
HIJ
• !"##$%&'()" )$##* +"!,*- $(."(/!*!$,"# cn =f (n) (a)
n!, )$ ,")" 0&$ 1")$,"# !$$#2
.!$3$! * #4!'$ ."," #$%&$
f (x) = f (a)+f ′ (a) (x− a)+f ′′ (a)
2!(x− a)2+
f (3) (a)
3!(x− a)3+· · ·+f (n) (a)
n!(x− a)n+· · ·
"& #$5*- $(."(/!*,"# * #4!'$ )$ 6*78"!9
f (x) =∞∑
n=0
f (n) (a)
n!(x− a)n .
!"#$% &'()'( !"!#$%&$!' !( ")'*! +! ,-.&%' - /0#12% f (x) = sin x.
!"#$%!& !',$'!" 3*,"# )$/$!,'(*! *# )$!'3*)*# )$ /")*# *# "!)$(# )$ f (x) = sin x !
"! #! a. $%&!' ()%
f (a) = sin a f ′ (a) = cos a f ′′ (a) = − sin af (3) (a) = − cos a f (4) (a) = sin a f (5) (a) = cos a
* '%+),-. ')/'#,#)0&!' 1 %2"-%''3! 41 '5-,% 4% $167!-
f (x) = f (a)+f ′ (a) (x− a)+f ′′ (a)
2!(x− a)2+
f (3) (a)
3!(x− a)3+ · · ·+ f (n) (a)
n!(x− a)n+ · · ·
% !/#%&!'
sin x = sin a+ cos a (x− a)− sin a
2!(x− a)2 − cos a
3!(x− a)3 +
sin a
4!(x− a)4 + · · · .
8'#1 '5-,% "!4% '%- -%%'9-,#1 '%"1-1 4! !' #%-&!' %& '% ! 4!' #%-&!' %& 9!''% !. 9! :
;!-&% '%+)%
sin x =
(
sin a− sin a
2!(x− a)2 +
sin a
4!(x− a)4 + · · ·
)
+(
cos a (x− a)− cos a
3!(x− a)3 + · · ·
)
,
% %'9-%<% 4! %& ;!-&1 4% '!&1#=-,! <%& ()%
sin x =∞∑
n=0
(−1)n sin a
2n!(x− a)2n +
∞∑
n=0
(−1)n cos a
(2n+ 1)!(x− a)2n+1
.
!"# $%&'( )( *+,-+.&'/
>!7, ?1971)-, @ABCD : AEFBG ;!, )& &1#%&H#,9! %'9!9I'J K1-1 !/#%- ! 4%'% <!7<,&% #!
4% )&1 ;) L3! %& '5-,% 4% ?1971)-, /1'#1 #!&1- a = 0 1 '5-,% 4% $167!-J M%''% &!4!. 1
'5-,% 4% ?19N1)-, 4% )&1 ;) L3! f 5 4141 "!-
f (x) =∞∑
n=0
fn (0)
n!xn = f (0) + f ′ (0) x+
f ′′ (0)
2!x2 +
f (3) (0)
3!x3 + · · ·+ f (n) (0)
n!xn + · · · .
!"#$% &'()'( !"!#$%&$!' !( ")'*! +! ,-.&-/'*# - 0/#12% f (x) = sin x.
!"#$%!& O! 82%&"7! PJABJA 4%'% <!7<%&!' f (x) = sin x %& '5-,% 4% $167!-J Q1R% 4! a = 0 %''% 4%'% <!7<,&% #!. !/#%&!'
STF
sin x =
(
sin 0− sin 0
2!(x− 0)2 +
sin 0
4!(x− 0)4 + · · ·
)
+
(
cos 0 (x− 0)− cos 0
3!(x− 0)3 + · · ·
)
! "#$%&
sin x = x− x3
3!+
x5
5!− x7
7!+
x9
9!+ · · ·
! %'()%&
sin x =∞∑
n=0
(−1)n x2n+1
(2n+ 1)!.
* +#', - . )#-/ 0#-'12%-& "#3 4-%()#" )'12!+)%)#"& 5!# '(,#-0%+ )# 2 (0#-46(2'% )#",%
"7-'# 7 , )% % -#,% -#%+& ! "#$%& #",% "7-'# 2 (0#-4# .%-% , ) 0%+ - -#%+ )# x.
8'()%& #",% "7-'# . )# "#- %.+'2%)% .%-% )#,#-3'(%- 0%+ - )# 2 (0#-46(2'% )# "7-'#"
(!37-'2%"9 : - #;#3.+ & "!<",',!'() x = π6(% "7-'# %2'3%& ,#3 " 5!#
π
6−
(π
6
)3
3!+
(π
6
)5
5!−
(π
6
)7
7!+
(π
6
)9
9!+ · · · = sin
π
6=
1
2.
!"#$% &'()'* !"!#$%&$!' !( ")'*! +! ,-./-0'*# - 10#23% f(x) =
∫
sin x
xdx.
!"#$%!& :-'3#'- )'0')'3 " 2%)% ,#-3 <,') ( =;#3.+ >9?@9? . - x, #(2 (,-%()
sin x
x= 1− x2
3!+
x4
5!− x6
7!+
x8
9!+ · · ·
8 "#4!'-& '(,#4-%3 " % "7-'# ,#-3 % ,#-3 # <,#3 "
∫
sin x
xdx =
∫
dx−∫
x2
3!dx+
∫
x4
5!dx−
∫
x6
7!dx+
∫
x8
9!dx+ · · ·
= x− x3
3!3+
x5
5!5− x7
7!7+
x9
9!9+ · · ·
=∞∑
n=0
(−1)n x2n+1
(2n+ 1)! (2n+ 1),
5!# 2 (0#-4# .%-% , ) 0%+ - -#%+ )# x.
!"#$% &'()'+ 45*&*6! ")'*!" +! 10#27!" 8-'- .-&.0&-' limx→0
sin x− x
x3.
!"#$%!& 8 .%-,'- )% "7-'# #(2 (,-%)% ( =;#3.+ >9?@9?& ,#3 " 5!#
sin x = x− x3
3!+
x5
5!− x7
7!+
x9
9!+ · · · (−1)n x2n+1
(2n+ 1)!+ · · ·
# #(,A
sin x− x = −x3
3!+
x5
5!− x7
7!+
x9
9!+ · · · (−1)n x2n+1
(2n+ 1)!+ · · · .
BC>
!"!#!$#% &'(%) %) *&#%) +%, x3, -$.%$/,&'%)
sin x− x
x3= − 1
3!+
x2
5!− x4
7!+
x6
9!+ · · · (−1)n x2n−2
(2n+ 1)!+ · · · .
0%,/&$/%
limx→0
sin x− x
x3= lim
x→0
(
− 1
3!+
x2
5!− x4
7!+
x6
9!+ · · · (−1)n x2n−2
(2n+ 1)!+ · · ·
)
= −1
6.
!"#$% &'()'* !"!#$%&$!' !( ")'*! +! ,-.&-/'*# - 0/#12% f(x) = sin(2x).
!"#$%!& 1$/-,!%,'-$/-2 "!'%) 34- & )5,!- #- 6&.7&4,!$ #- sin x 5
sin x = x− x3
3!+
x5
5!− x7
7!+ · · · (−1)n x2n+1
(2n+ 1)!+ · · ·
/,%.&$#% x +%, 2x $-)/& )5,!-2 %(/-'%)
sin(2x) = 2x− (2x)3
3!+
(2x)5
5!− (2x)7
7!+ · · · (−1)n (2x)2n+1
(2n+ 1)!+ · · ·
= 2x− 23x3
3!+
25x5
5!− 27x7
7!+ · · ·+ (−1)n2
2n+1x2n+1
2n+ 1+ · · ·
=∞∑
n=0
(−1)n22n+1(x)2n+1
(2n+ 1)!.
8'& #&) +,!$.!+&!) &+*!.&9:-) #&) )5,!-) #- ;&<*%, - #- 6&.7&4,!$ %.%,,- $& !$/-=,&9>%
#- ?4$9:-)@ A-B/%$ ?,-34-$/-'-$/- !$/-=,&"& ?4$9:-) -C+,-))&$#%D&) +,!'-!,% .%'% 4'&
)5,!- #- +%/E$.!&) - #-+%!) !$/-=,&$#% & )5,!- /-,'% & /-,'%@
0%, -C-'+*%2 & ?4$9>% g(x) = e−x2
$>% +%#- )-, !$/-=,&#& +-*&) /5.$!.&) #% FG*.4*% H2
+%!) )4& &$/!#-,!"&#& $>% 5 4'& ?4$9>% -*-'-$/&,@ A% -C-'+*% & )-=4!, 4)&,-'%) & !#-!& #-
A-B/%$ +&,& !$/-=,&, -))& ?4$9>%@
!"#$% &'()'& 345'!""!
∫
e−x2
dx .%(% /(- ")'*! +! 5%67#.*-" .!#6'-+- #% 5%#6% 89
!"#$%!& 0,!'-!,% -$.%$/,&,-'%) & )5,!- #- 6&.7&4,!$ +&,& g(x) = e−x2
. I'(%,& )-J&
+%))K"-* 4)&, % '5/%#% #!,-/%2 "&'%) -$.%$/,GD*& & +&,/!, #& )5,!- #- 6&.7&4,!$ +&,& f(x) =ex. F%'% f (n)(x) = ex +&,& /%#% n $&/4,&*2 /-'%) 34-
f (n)(0) = e0 = 1 ∀n ∈ N∗
- &))!'2 & )5,!- #- 6&.7&4,!$ #& ?4$9>% -C+%$-$.!&* 5
ex =∞∑
n=0
f (n)(0)
n!xn =
∞∑
n=0
xn
n!= 1 + x+
x2
2!+
x3
3!+ · · · .
0%#-D)- '%)/,&, ?&.!*'-$/- 34- -)/& )5,!- .%$"-,=- +&,& /%#% x ,-&* - 34- )-4 !$/-,"&*%
#- .%$"-,=E$.!& 5 !$L$!/%@ ;,%.&$#% x +%, −x2$-)/- #-)-$"%*"!'-$/%2 %(/-'%) 34-
e−x2
=∞∑
n=0
(−x2)n
n!=
∞∑
n=0
(−1)nx2n
n!= 1− x2 +
x4
2!− x6
3!+ · · ·
MNO
!" #$%&'% ()*+",-" .$,$ #)/) x. 0-),$ .)/"%)1 2*#"-,$, "1#$ 1',2" #",%) $ #",%)3 /"
$(),/) ()% ) 4"),"%$ 567567 " )&#", ∀n ∈ R
∫
e−x2
dx = C +∞∑
n=0
(−1)nx2n+1
(2n+ 1)n!= C + x− x3
3+
x5
5.2!− x7
7.3!+ · · ·
!"#$% &'()'* !"#$"%
∫ 1
0
e−x2
dx #&' $'! ()%#*+,& -% .)/+ #!+!+ -%#*'!*+0
!"#$%!& 0.82($*/) ) 4"),"%$ 9!*/$%"*#$8 /) :;8(!8) < "=.,"11>) )/$ *) "="%.8)
$*#",2),3 #"%)1 !"
∫ 1
0
e−x2
dx = C +∞∑
n=0
(−1)nx2n+1
(2n+ 1)n!
∣
∣
∣
∣
∣
1
0
=∞∑
n=0
(−1)n(2n+ 1)n!
.
?=.$*/2*/) $8-!*1 #",%)1 /"1#$ 1',2" *!%',2($3 #"%)1 !"
∫ 1
0
e−x2
dx =∞∑
n=0
(−1)n(2n+ 1)n!
= 1− 1
3
1
10− 1
42+
1
216− 1
1320+
1
9360+ · · ·
" )&1",+$%)1 !" $ .$,#2, /) 1"=#) #",%) /"1#$ "=.$*1>)3 #)/)1 )1 /"%$21 .)11!"% %@/!8)
%"*), !"
11320
< 0, 001 " $112%3 $) 1)%$,%)1 )1 (2*() .,2%"2,)1 #",%)1 /$ "=.$*1>) #","%)1
!%$ $.,)=2%$A>) ()% .,"(21>) /" $#' 3 ($1$ /"(2%$21
∫ 1
0
e−x2
dx ≈ 1− 1
3+
1
10− 1
42+
1
216≈ 0, 7475.
!"#$% &'()') 1.*"*2% -%+%34&"4*'%3.& %' +5)*%+ -% 6!#7!$)*3 (!)! #!"#$"!)
limx→0
arctan(x)− sin x
x3 cos x.
!"#$%!& :)%"A$%)1 ()% ) /"1"*+)8+2%"*#) "% 1',2" /" .)#B*(2$1 /" f(x) = arctan x.:)%)
f ′(x) =1
1 + x2= (1 + x2)−1
' %$21 12%.8"1 2*2(2$, ."8) /"1"*+)8+2%"*#) /" f ′. C) ?="%.8) 567D67 )&#"%)1 !"
(1 + x)−1 = 1− x+ x2 − x3 + x4 + · · ·+ (−1)nxn + · · ·
#,)($*/) x .), x2, 1"-!" !"
f ′(x) = (1 + x2)−1 = 1− x2 + x4 − x6 + · · ·+ (−1)nx2n + · · ·
"*#>)3 2*#"-,$*/) #",%) $ #",%)3 #"%)1 !"
arctanx =
∫
1
1 + x2dx = x− x3
3+
x5
5− x7
7+ · · ·+ (−1)nx2n+1
2n+ 1+ · · · 'I)
E$ ()*1#$*#" *$ "=.$*1>) /$ F!*A>) $,() #$*-"*#" ' G",)H6
02*/$3 1$&"%)1 !" ) /"1"*+)8+2%"*#) "% 1',2" .$,$ ) 1"*) '
sin x = x− x3
3!+
x5
5!− x7
7!+ · · ·+ (−1)nx2n+1
(2n+ 1)!+ · · · 'II)
IJK
!"#$%! # %&'()($*# ($+)( #, (-.#*/(, !" ( !!" !0+("!,
arctanx− sin x = x3
(−13
+1
3!
)
+ x5
(
1
5− 1
5!
)
+ · · ·+ x2n+1
(
(−1)n2n+ 1
+(−1)n+1
(2n+ 1)!
)
+ · · ·
1!%("!, !0+() # ,2)&( %( 3#45#.)&$ 6#)# cosx '#4&7"($+(8 0#,+# %()&9#) +()"! # +()"!# ,2)&( %( sin x %(,($9!79&%# #4&"#8 !0+($%!
cosx = 1− x2
2!+
x4
4!− x6
6!+ · · ·+ (−1)n x2n
(2n)!+ · · · .
:;!)# 6!%("!, +!"#) ! -.!4&($+( %(,(<#%! ( ,&"67&=4#)8 6#)# !0+() -.(
arctan(x)− sin x
x3 cos x=
x3
(−13
+1
3!
)
+ x5
(
1
5− 1
5!
)
+ · · ·+ x2n+1
(
(−1)n2n+ 1
+(−1)n+1
(2n+ 1)!
)
+ · · ·
x3
(
1− x2
2!+
x4
4!+ · · ·+ (−1)nx2n
(2n)!+ · · ·
)
=
(−13
+1
3!
)
+ x2
(
1
5− 1
5!
)
+ · · ·+ x2n−2
(
(−1)n2n+ 1
+(−1)n+1
(2n+ 1)!
)
+ · · ·(
1− x2
2!+
x4
4!− x6
6!+ · · ·+ (−1)n x2n
(2n)!+ · · ·
)
>&$#7"($+(8 6!%("!, #67&4#) ! 7&"&+( (" #"0!, !, 7#%!, %(,,# &;.#7%#%( ( ($4!$+)#) -.(
limx→0
arctan(x)− sin x
x3 cosx=
(−13
+1
3!
)
+ 0
1 + 0=−13
+1
6= −1
6.
!"# $%&'()* +,&*) -. /012'0. -, 3,45.1
?.6!$@#"!, -.( ! &$+()(,,( 2 ! %(,($9!79&"($+! %! 0&$A"&! (a+ b)n , 6#)# n &$+(&)!
6!,&+&9!B C! %(,($9!79&"($+! ;()#7 %! 0&$A"&$! %( D(E+!$ 9(" -.(
(a+ b)n = C0na
n + C1na
n−1b+ C2na
n−2b2 + · · ·+ Ckna
n−kbk + · · ·+ Cnnb
n.
F!"!
Ckn =
n!
k! (n− k)!=
n (n− 1) (n− 2) · · · (n− (k − 1)) (n− k)!
k! (n− k)!=
n (n− 1) (n− 2) · · · (n− (k − 1))
k!,
6!%("!, (,4)(9()
(a+ b)n = an+nan−1b+n (n− 1)
2!an−2b2+· · ·+n (n− 1) (n− 2) · · · (n− (k − 1))
k!an−kbk+· · ·+bn.
!"#$%! a = 1 ( b = x 9(" -.(
(1 + x)n = 1 + nx+n (n− 1)
2!x2 + · · ·+ n (n− 1) (n− 2) · · · (n− (k − 1))
k!xk + · · ·+ xn,
-.( 2 ." %(,($9!79&"($+! =$&+!B 1!)2"8 ,( n $G! '!) ." &$+(&)! 6!,&+&9! !. H()!8 2 4!$I
9($&($+( %(,($9!79() ! 0&$A"&! (1 + x)n (" ,2)&( %( 3#47#.)&$B C(,,( "!%! +()("!, !
%(,($9!79&"($+! &$=$&+!
(1 + x)n = 1 + nx+n (n− 1)
2!x2 +
n (n− 1) (n− 2)
3!x3 + · · ·+
+n (n− 1) (n− 2) · · · (n− k + 1)
k!xk + · · · JKBLMBLN
OPM
!"# !$%&'( )*#+#,# ,' !$%&' -&./+�( $ 1+ )#!/ 2#%"&)10#% ,# 3$%&' ,' 4#)5#1%&.6 7/+/
/ 0'&"/% 2/,'%8 9'%&:)#%( #"%#9$! ,/ 7%&"$%&/ ,' ;<=0'+-'%"( # !$%&' -&./+� $ #-!/01"#+'."'
)/.9'%>'."' 2#%# "/,/ x %'#0 "#0 ?1' |x| < 1. @/,' !'% 2%/9#,/ ?1' '!!' ,'!'.9/09&+'."/$ 9'%,#,'&%/ 2#%# "/,/ n. = 2%/9# 2/,' !'% '.)/."%#,# ./! 0&9%/! )&"#,/! .# -&-0&/>%#:#6
!)%'9'.,/ '+ A/%+# ,' !/+#"B%&/( "'+/! ?1'
(1 + x)n = 1 +∞∑
k=1
n (n− 1) (n− 2) · · · (n− k + 1)
k!xk
!' |x| < 1.
!"#$% &'()'( !"!#$%&$!' !( ")'*! +! ,-#./!" 0 ,-#.1% f (x) =1
1 + x.
!"#$%!& C'+/! ?1'
f (x) =1
1 + x= (1 + x)−1 .
@/%"#."/( -#!"# !1-!"&"1&% n = −1 .# AB%+10# ,# !$%&' -&./+ =!!&+(
1
1 + x= 1 + (−1)x+
−1 (−1− 1)
2!x2 +
−1 (−1− 1) (−1− 2)
3!x3 + · · ·
+−1 (−1− 1) (−1− 2) · · · (−1− k + 1)
k!xk + · · ·
= 1− x+2
2!x2 +
−63!
x3 + · · ·+ −1 (−1− 1) (−1− 2) · · · (−1− k + 1)
k!xk + · · ·
1
1 + x= 1− x+ x2 − x3 + x4 + · · ·+ (−1)kxk + · · · =
∞∑
k=0
(−1)k xk.
!"#$% &'()'* 234'!""! 5%(% -(0 ")'*! +! 4%67#5*0" 0 ,-#.1% f(x) =ln(x+ 1)
x.
!"#$%!& D#+/! #.#0&!#% &.&)�+'."' # A1.EF/ ln(x+ 1). = !1# ,'%&9#,# $ &>1#0 #
1
x+ 1, '
./ 'G'+20/ #."'%&/% +/!"%#+/! ?1'
1
x+ 1= 1− x+ x2 − x3 + x4 + · · ·+ (−1)nxn + · · · =
∞∑
n=0
(−1)n xn,
2/%"#."/( ,'9'+/! &."'>%#% #+-/! /! +'+-%/! ,# &>1#0,#,'( /-"'.,/
ln(x+ 1) =
∫
1
1 + xdx =
∞∑
n=0
∫
(−1)n xndx =∞∑
n=0
(−1)n xn+1
n+ 1.
7/+/ ?1'%'+/! f(x) =ln(x+ 1)
x, ,'9'+/! ,&9&,&% "/,/! /! +'+-%/! 2/% x, ,/.,'(
ln(x+ 1)
x=
∞∑
n=0
(−1)n xn
n+ 1.
!"#$% &'()'+ !"!#$%&$!' !( ")'*! +! ,-#./!" 0 ,-#.1% f (x) =1√1 + x
.
!"#$%!& C'+/! ?1'
HIJ
f (x) =1√1 + x
= (1 + x)−1
2 .
!"#$%#!& '$(#$ ()'(#*#)*" n = −12%$ +,"-).$ /$ (0"*1 '*%!-*$.2 3((*-&
1√1 + x
= 1 +
(
−1
2
)
x+−1
2
(
−12− 1
)
2!x2 +
−12
(
−12− 1
) (
−12− 2
)
3!x3 + · · ·
+−1
2
(
−12− 1
) (
−12− 2
)
· · · (−12− k + 1)
k!xk + · · ·
= 1− 1
2x+
−1
2
(
−3
2
)
2!x2 +
−1
2
(
−3
2
)(
−5
2
)
3!x3 + · · ·
+
−1
2
(
−3
2
)(
−5
2
)
· · · (1− 2k
2)
k!xk + · · ·
1√1 + x
= 1− 1
2x+
1 · 3222!
x2 − 1 · 3 · 5233!
x3 + · · ·+ (−1)k 1 · 3 · 5 · ... · (2k − 1)
2kk!xk + · · ·
!"#$% &'()'* !"!#$%&$!' !( ")'*! +! ,-#./!" 0 ,-#.1% f (x) =1√
1− x2.
!"#$%!& !/1-!( $4"!51*#$" ! "1().#$/! /! 671-4.! 829:2; ()'(#*#)*%/! x 4!" (−x2) .<1"1-!( 1%#=!
1√
1 + (−x2)= 1− 1
2
(
−x2)
+1 · 3222!
(
−x2)2 − 1 · 3 · 5
233!
(
−x2)3
+ · · ·
+(−1)n 1 · 3 · 5 · · · (2n− 1)
2nn!
(
−x2)n
+ · · ·1√
1− x2= 1 +
1
2x2 +
1 · 3222!
x4 +1 · 3 · 5233!
x6 + · · ·+ 1 · 3 · 5 · ... · (2n− 1)
2nn!x2n + · · ·
!"#$% &'()'& !"!#$%&$!' !( ")'*!" +! ,-#./!" 0 ,-#.1% f (x) = arcsin x.
!"#$%!& >!-! $ /1"*5$/$ /$ +)%?=! f (x) = arcsin x 0 f ′ (x) =1√
1− x24!/1-!(
$4"!51*#$" ! "1().#$/! /! 671-4.! 829:2@ 1 *%#1A"BC.! #1"-! $ #1"-!& !'#1%/!
∫
dx√1− x2
=
∫
dx+1
2
∫
x2dx+1 · 3222!
∫
x4dx+1 · 3 · 5233!
∫
x6dx+ · · ·
+1 · 3 · 5 · ... · (2n− 1)
2nn!
∫
x2ndx+ · · ·
D)1 "1().#$ 1-
arcsin x = x+1
2 · 3x3 +
1 · 3222!5
x5 +1 · 3 · 5233!7
x7 + · · ·+ 1 · 3 · 5 · ... · (2n− 1)
2nn! (2n+ 1)x2n+1 + · · ·+ C
!) (1E$
arcsin x = x+∞∑
n=1
1 · 3 · 5 · ... · (2n− 1)
2nn! (2n+ 1)x2n+1 +
π
2.
!+,!-./01% &'()'2 20&! '!""0&30' 4-! % +!"!#$%&$*(!#3% %53*+% !( 3%+%" %" !6!(7&%" 0#3!8
'*%'!" ) $9&*+% 07!#0" 70'0 |x| < 1.
F9G
!"# $%&'()(*+, -&'.*,
! "#$#%&'(# )* +,-$%) .%'&#'%)* $#%&)* /# 0-/- ,&- /-* *#+,1(0'-* /-/-* -2-'3)! 4-56
0,5# $-&27& limn→∞
un, 0-*) #3'*$-!
(a) un = n4n+2
(b) un = (−1)n
5−n (c) un = (−1)n√n
n+1(d) un = 100n
n32+4
(e) un = n+1√n
(f) un = lnnn
(g) un = ln(
1n
)
(h) un = n2
5n+3
(i) un = cos nπ2
(j) un = arctann (k) un =(
1− 2n
)n(l) un = n2
2n
(m) un = 3ne2n
(n) un = 1 + (−1)n (o) un = n√n (p) un = 7−n3n−1
8! "-/)* )* $#%&)* -2-'3)9 /#$#%&'(# ,&- #3.%#**:) .-%- -* *#+,1(0'-*!
(a){
13, 29, 427, 881, · · ·
}
(b){
13, −2
9, 427, −881, · · ·
}
(c){
12, 34, 56, 78, · · ·
}
(d){
0, 14, 29, 316, · · ·
}
;! 45-**'<+,#9 *# .)**=>#59 -* *#+,1(0'-* -2-'3) +,-($) ? *,- &)()$)('0'/-/#!
(a) un = n2n−1 (b) un = n− 2n (c) un = ne−n (d) un = 5n
2n2
(e) un = 10n
(2n)!(f) un = nn
n!(g) un = 1
n+lnn(h) un = n!
3n
@! A,.)(B- +,# un *#C- ,&- *#+,1(0'- &)(D$)(- $-5 +,# 1 ≤ un ≤ 5. E*$- *#+,1(0'-
/#># 0)(>#%F'%G H +,# &-'* .)/# *#% /'$) *)2%# ) *#, 5'&'$#G
I! A,.)(B- +,# un *#C- ,&- *#+,1(0'- &)(D$)(- $-5 +,# un ≤ 5. E*$- *#+,1(0'- /#>#
0)(>#%F'%G H +,# &-'* .)/# *#% /'$) *)2%# ) *#, 5'&'$#G
J! K)/#6*# )2$#% -.%)3'&-LM#* /#
√k ,$'5'N-(/) - *#+,1(0'- %#0,%*'>- un+1 =
12
(
un +kun
)
,
)(/# u1 =12.
O-P E(0)($%# -* -.%)3'&-LM#* u2, u3, u4, u5, u6 .-%-
√10.
O2P Q)*$%# +,#9 *# L = limn→∞
un, #($:) L =√k.
R! S&- /-* &-'* T-&)*-* *#+,1(0'-* 7 - *#+,1(0'- /# U'2)(-00' O R V6 8IVP9 /#<('/-
.#5- %#0)%%1(0'- un+1 = un + un−1, )(/# u1 = u2 = 1.
O-P "#$#%&'(# )* /#N .%'&#'%)* $#%&)* /#*$- *#+,1(0'-!
O2P H* $#%&)* /- ()>- *#+,1(0'- xn = un+1
un/:) ,&- -.%)3'&-L:) .-%- ) 'F,-5&#($#
T-&)*) (W&#%) /# ),%) O), %-N:) X,%#-P9 /#()$-/) .)% τ. "#$#%&'(# ,&- -.%)3'&-L:)
/)* 0'(0) .%'&#'%)* $#%&)* /#**- ()>- *#+,1(0'-!
O0P A,.)(/) +,# τ = limn→∞
xn, &)*$%# +,# τ = 12(1 +
√5).
Y! E(0)($%# ) $#%&) F#%-5 /- *#+,1(0'- /# *)&-* .-%0'-'* /# 0-/- ,&- /-* *7%'#* -2-'3)!
Z *#F,'%9 /#$#%&'(# *# - *7%'# 0)(>#%F# ), /'>#%F#9 )2$#(/) ) >-5)% /# *,- *)&-9 *#
.)**=>#5!
8
(a)∞∑
n=1
1
(2n− 1) (2n+ 1)(b)
∞∑
n=1
8
(4n− 3) (4n+ 1)
(c)∞∑
n=1
2n+ 1
n2 (n+ 1)2(d)
∞∑
n=1
ln
(
n
n+ 1
)
(e)∞∑
n=1
2n−1
5n(f)
∞∑
n=1
1√
n (n+ 1)(√
n+ 1 +√n)
(g)∞∑
n=1
1
1.2.3.4.5. · · · .n.(n+ 2)(h)
∞∑
n=1
3n+ 4
n3 + 3n2 + 2n
! "#$%&'( '( $' $)*+$,-(' $.$&/0 '10 2(*3$3(&*$' 04 5$%'$'! 64'7&)84( '(4' $*94+(#:
70'; (/&. <0#7*$:(/(+=%0' =$*$ $' $)*+$,-(' 5$%'$' 04 =*02$#30 $' $)*+$,-('
2(*3$3(&*$'!
>$? @03$ '(84A#<&$ %&+&7$3$ B <0#2(*9(#7(!
>.? @03$ '(84A#<&$ %&+&7$3$ B +0#C70#$!
><? @03$ '(84A#<&$ <0#2(*9(#7( B #(<(''$*&$+(#7( +0#C70#$!
>3? @03$ '(84A#<&$ +0#C70#$ 3(<*('<(#7( <0#2(*9( =$*$ D(*0!
>(? E( un 50* 3(<*('<(#7( ( un > 0 =$*$ 7030 n ∈ N, (#710 un B <0#2(*9(#7(!
>5? E( −1 < q < 1, (#710 limn→+∞
qn = 0.
>9? E( $ '(84A#<&$ un <0#2(*9(; (#710 $ 'B*&(
∞∑
n=1
un 7$+.B+ <0#2(*9(!
>F? E(
∞∑
n=1
un <0#2(*9(; (#710
∞∑
n=1
√un 7$+.B+ <0#2(*9(!
>&? @03$ 'B*&( $%7(*#$3$ <0#2(*9(#7( B <0#3&<&0#$%+(#7( <0#2(*9(#7(!
>G? " 'B*&(
∞∑
n=1
(n3 + 1)2
(n4 + 5)(n2 + 1)B 4+$ 'B*&( #4+B*&<$ <0#2(*9(#7(!
>H? I('(#20%2(#30 $ 54#,10 g(x) =
∫ x
0
t2e−t2
dt (+ 'B*&( 3( =07A#<&$' 0.7B+:'( g(x) =
∞∑
n=0
(−1)nx2n+3
n!(2n+ 3).
>%? " 'B*&( 3( =07A#<&$'
∞∑
n=1
(−1)3nxnB <0#2(*9(#7( #0 (*2$%0 (−1
3, 13) ( '4$ '0+$ B
&94$% $ S =−3x1 + 3x
.
>+? E( $ '(84A#<&$ un <0#2(*9( (#710 $ 'B*&(
∞∑
n=1
(un+1 − un) 7$+.B+ <0#2(*9(!
>#? J *$&0 3( <0#2(*9A#<&$ 3$ 'B*&( 3$ 'B*&(
∞∑
n=0
(−1)n(3x− 5)2n
22n(n!)2B &#)#&70!
>0? " 'B*&(
∞∑
n=1
22n91−n B <0#2(*9(#7( ( '4$ '0+$ B &94$% $
36
5.
>=? J <*&7B*&0 3$ (9*$% 9$*$#7( 84(
∞∑
n=3
1
n lnn ln(lnn)<0#2(*9(!
KLK
!" #$%&$'() & ')(*& +)(,- ., /&*, ., /0(1)
∞∑
n=1
4
4n2 − 1) 2)(1345) /) )-, 0 %&$2)(+)$')"
" #$%&$'() , /&*, .,/ /0(1)/ ,6,17&8 /) 9&//:2)-"
(a)∞∑
n=1
(
1
5
)n
(b)∞∑
n=1
5
(5n+ 2)(5n+ 7)(c)
∞∑
n=1
1
n2 + 6n+ 8(d)
∞∑
n=1
−1√n+ 1 +
√n
;" </,$.& & ')/') .) %&*9,(,=>& 2)(1345) /) ,/ /0(1)/ ,6,17& />& %&$2)(+)$')/ &5 .12)(?
+)$')/"
(a)∞∑
n=1
1
n3n(b)
∞∑
n=1
√n
n2 + 1(c)
∞∑
n=1
1
nn(d)
∞∑
n=1
n2
4n3 + 1
(e)∞∑
n=1
1√n2 + 4n
(f)∞∑
n=1
|sen(n)|2n
(g)∞∑
n=1
n!
(2 + n)!(h)
∞∑
n=1
1√n3 + 5
(i)∞∑
n=1
1
n√n2 + 5
(j)∞∑
n=1
1
n+√n+ 5
(k)∞∑
n=1
n
4n3 + n+ 1(l)
∞∑
n=1
2n
(2n)!
(m)∞∑
n=1
√n+ 1 +
√n
3√n
(n)∞∑
n=1
1 + n42n
n5n(o)
∞∑
n=1
2 + cosn
n2(p)
∞∑
n=1
√n
n+ 4
(q)∞∑
n=1
1 + 2n
1 + 3n(r)
∞∑
n=1
n+ lnn
n3 + 1
@" </,$.& & ')/') .) A BC-,*6)(' 2)(1345) /) ,/ /0(1)/ ,6,17& />& %&$2)(+)$')/ &5 .12)(?
+)$')/"
(a)∞∑
n=1
n+ 1
n22n(b)
∞∑
n=1
n!
en(c)
∞∑
n=1
1
(n+ 1)2n+1
(d)∞∑
n=1
3n√n3 + 1
(e)∞∑
n=1
3n
2n(n2 + 2)(f)
∞∑
n=1
n!
2n (2 + n)!
(g)∞∑
n=1
1
n+ 5(h)
∞∑
n=1
n+ 1
n4n(i)
∞∑
n=1
n
4n2 + n+ 1
(j)∞∑
n=1
3n+ 1
2n(k)
∞∑
n=1
3n
n2 + 2(l)
∞∑
n=1
n!
(n+ 2)3(m)
∞∑
n=1
2n−1
5n(n+ 1)
D" </,$.& & ')/') .) E,5%FG8 2)(1345) /) ,/ /0(1)/ ,6,17& />& %&$2)(+)$')/ &5 .12)(+)$')/"
(a)∞∑
n=1
(lnn)
nn
2
n
(b)∞∑
n=1
2n(
n+ 1
n2
)n
(c)∞∑
n=1
(
n+ 1
n22n
)n
(d)∞∑
n=1
n4n − n√n10n + 1
!" #$%&'( ( )*$)* '% +&)*,-%. /*-+012* $* %$ $3-+*$ %4%+5( $6( 7(&/*-,*&)*$ (2 '+/*-,*&)*$"
(a)∞∑
n=1
ne−n (b)∞∑
n=1
lnn
n(c)
∞∑
n=2
1
n lnn(d)
∞∑
n=1
1
(n+ 1)√
ln (n+ 1)
(e)∞∑
n=1
arctann
n2 + 1(f)
∞∑
n=1
ne−n2
(g)∞∑
n=1
n2e−n (h)∞∑
n=1
earctann
n2 + 1
(i)∞∑
n=1
1
4n+ 7(j)
∞∑
n=1
1
n√n2 + 1
(k)∞∑
n=1
1
n(1 + ln2 n)
8" 9*-+012* $* %$ $3-+*$ %4%+5( $6( %4$(.2)%:*&)* 7(&/*-,*&)*; 7(&'+7+(&%.:*&)* 7(&/*-<
,*&)* (2 '+/*-,*&)*"
= >
(a)∞∑
n=1
(−1)n−1 2n
n!(b)
∞∑
n=1
(−1)n−1 1
(2n− 1)!(c)
∞∑
n=1
(−1)n−1 n2
n!
(d)∞∑
n=1
(−1)n−1 n(
2
3
)n
(e)∞∑
n=1
(−1)n−1 n!
2n+1(f)
∞∑
n=1
(−1)n−1 1
n2 + 2n
(g)∞∑
n=1
(−1)n−1 3n
n!(h)
∞∑
n=1
(−1)n−1 n2 + 1
n3(i)
∞∑
n=1
(−1)n−1 nn
n!
(j)∞∑
n=1
(−1)n−1 1
n2
3 + n(k)
∞∑
n=1
(−1)n−1 nn2n
(2n− 5)n(l)
∞∑
n=1
(−1)n−1n4
en
(m)∞∑
n=1
(−1)n−1 n
n2 + 1(n)
∞∑
n=1
(−1)n−1 n
n3 + 3(o)
∞∑
n=1
(−1)n√2n2 − n
!" #$%&&'()*+ %& &,-'+& .*/,-'0%& %1%'23 03/3 %1&3$*4%/+.4+ 03.5+-6+.4+7 03.8'0'3.%$9
/+.4+ 03.5+-6+.4+ 3* 8'5+-6+.4+7 :*&4'(0%.83 &*% -+&;3&4%"
(a)∞∑
n=1
(−1)n−1 (23n+4 − n)
enn3n(b)
∞∑
n=1
n cos(nπ)
n2 + n+ 1(c)
∞∑
n=1
(−1)n√
n+√n
(d)∞∑
n=1
(−1)n(n+ 1)!
2.4.6 · · · .(2n) (e)∞∑
n=1
(−1)n 54n+1
n3n(f)
∞∑
n=1
(−1)n 73n+1
(lnn)n
(g)∞∑
n=1
n sin(nπ) + n
n2 + 5(h)
∞∑
n=1
cos(n) + sin(n)
n3 +√n
(i)∞∑
n=1
ne2n
n2en − 1
<" =+4+-/'.+ 3 -%'3 + 3 '.4+-5%$3 8+ 03.5+-6>.0'% 8%& &,-'+& 8+ ;34>.0'%& %1%'23"
(a)∞∑
n=1
xn
√n
(b)∞∑
n=1
(−1)n−1xn
n3(c)
∞∑
n=0
(3x− 2)n
n!
(d)∞∑
n=1
(−1)nn4nxn (e)∞∑
n=1
(−2)nxn
4√n
(f)∞∑
n=2
(−1)nxn
4n lnn
(g)∞∑
n=0
n(x+ 2)n
3n+1(h)
∞∑
n=0
√n(x− 4)n (i)
∞∑
n=1
(−1)n(x+ 2)n
n2n
(j)∞∑
n=1
n!(2x− 1)n (k)∞∑
n=1
xn
n√n3n
(l)∞∑
n=1
(4x− 5)2n+1
n3
2
(m)∞∑
n=0
n(x− 5)n
n2 + 1(n)
∞∑
n=0
nn(x+ 2)n
(2n− 5)n(o)
∞∑
n=0
n4(x− 1)n
en
(p)∞∑
n=0
2n(x+ 1)n
n2 + 1(q)
∞∑
n=0
n(x− 1)2n
n3 + 3(r)
∞∑
n=1
(−1)n1.3.5.7. · · · .(2n− 1)xn
3.6.9. · · · .3n
?" @+:% f(x) =∞∑
n=1
xn
n2. =+4+-/'.+ 3& '.4+-5%$3& 8+ 03.5+-6>.0'% ;%-% f, f ′ + f”.
AB" C ;%-4'- 8% &3/% 8% &,-'+ 6+3/,4-'0%
∞∑
n=1
xn, ;%-% |x| < 1, +.03.4-+ %& &3/%& 8%& &,-'+&
A D
! "#$%
(a)∞∑
n=1
nxn−1 (b)∞∑
n=1
nxn (c)∞∑
n=1
n
2n(d)
∞∑
n=2
n(n− 1)xn
(e)∞∑
n=2
n2 − n
2n(f)
∞∑
n=1
n2
2n(g)
∞∑
n=1
(−1)nxn
n(h)
∞∑
n=0
(−1)n2n(n+ 1)
&'% ()*$)+,- ./ ,-0,-1-)+ 23$ -/ 14,"- 5- 0$+6)*" 17 *-)+, 5 1 -/ 8-,$7 0 , 1 9.)2:-1
! "#$%
(a) f(x) =1
1 + x3(b) f(x) =
1
4 + x3(c) f(x) =
x
9 + 4x2
(d) f(x) =x2
(1− 2x)2(e) f(x) =
x3
(x− 2)2(f) f(x) = ln(5− x) (g) f(x) = x ln(x2 + 1)
&&% (#0,-11- 1 ")+-;, "1 ")5-<)"5 1 ! "#$ *$/$ ./ 14,"- 5- 0$+6)*" 17 *-)+, 5 1 -/
8-,$%
(a)
∫
x
1− x8dx (b)
∫
ln(1− x2)
x2dx (c)
∫
x− arctanx
x3dx (d)
∫
arctanx2dx
&=% >+"?"8- ,-0,-1-)+ 23$ -/ 14,"- 5- 0$+6)*" 17 *-)+, 5 -/ 8-,$7 5- f(x) = arctanx0 , 0,$@ , 1-;.")+- -#0,-113$ 0 , π *$/$ 1$/ 5- ./ 14,"- )./4,"* A π =
2√3∞∑
n=0
(−1)n3n(2n+ 1)
.
&B% C$1+,- D.- 9.)23$ f(x) =∞∑
n=0
xn
n!4 1$?.23$ 5 -D. 23$ 5"9-,-)*" ? f ′(x) = f(x).
&E% C$1+,- D.- 1 9.)2:-1 f1(x) =∞∑
n=0
(−1)nx2n
(2n)!- f2(x) =
∞∑
n=0
(−1)nx2n+1
(2n+ 1)!13$ 1$?.2:-1
5 -D. 23$ 5"9-,-)*" ? f”(x) + f(x) = 0.
&F% ()*$)+,- 1$/ 5 1 1-;.")+-1 14,"-1
(a)∞∑
n=0
(−1)nπ2n+1
42n+1(2n+ 1)!(b)
∞∑
n=0
(−1)nπ2n
62n(2n)!(c)
∞∑
n=1
3n
n!(d)
∞∑
n=0
3n
5nn!
&G% ()*$)+,- $ , "$ - $ 5$/H)"$ 5- *$)@-,;6)*" 5 14,"-
∞∑
n=0
2n(x− 2)n
5n(1 + n2).
&I% J-+-,/")- $ ")+-,@ ?$ 5- *$)@-,;6)*" 5 14,"-
∞∑
n=1
(3x− 5)n
7nn.
&K% C$1+,- D.- 14,"- 5- 0$+6)*" 1
∞∑
n=0
(−1)nx2n
32n4 *$)@-,;-)+- )$ ")+-,@ ?$ (−3, 3) - D.-
1. 1$/ 4 ";. ? S =9
9 + x2.
=L% J-+-,/")- $ ")+-,@ ?$ 5- *$)@-,;6)*" 5 14,"- 5- 0$+6)*" 1 D.- ,-0,-1-)+ 9.)23$
f(x) =4
x2-#0 )5"5 -/ +$,)$ 5- a = 1.
='% J-1-)@$?@ 9.)23$ f(x) = cosh(x3) -/ 14,"- 5- C *M .,")7 5-+-,/") )5$ $ +-,/$
;-, ? 5- 1. -#0 )13$ - $ 1-. ")+-,@ ?$ 5- *$)@-,;6)*" %
&'E
!" #$%$&'()$ * ()%$&+,-* $ * &,(* .$ /*)+$&01)/(, ., 23&($ .$ 45)67$28 /$)%&,., $' 9$&*8
:5$ &$;&$2$)%, , 45)6<* f(x) =ex
2 − 1
x.
" =2,).* 23&($2 .$ >,/-,5&()8 '*2%&$ :5$
∫
cosxdx = sin x+ k.
?" #$2$)+*-+, , 45)6<* f(x) =
∫ x
0
t2 ln(1 + 4t2)dt $' 23&($2 .$ >,/@,5&() $ .$%$&'()$ *
2$5 ()%$&+,-* .$ /*)+$&01)/(,"
A" #$2$)+*-+$& $' 23&($ .$ B,C-*& $ >,/-,5&() ,2 45)67$2D
(a) f(x) = sin2 x (b) f(x) = x2 sin 2x (c) f(x) = e3x (d) f(x) = e−x2
(e) f(x) = cos 2x (f) f(x) =sin(x5)
x3(g) f(x) =
cosx− 1
x2(h) f(x) = x3ex
2
E" =%(-(9$ .$2$)+*-+('$)%* $' 23&($2 .$ >,/@,5&() ;,&, /,-/5-,& *2 2$05()%$2 -('(%$2"
(a) limx→0
cos 2x+ 2x2 − 1
x4(b) lim
x→0
sin(x2) + cos(x3)− x2 − 1
x6
(c) limx→0
ln(1 + x2)
1− cosx(d) lim
x→0
ln(1 + x2)− 3 sin(2x2)
x2
(e) limx→0
ln(1 + x3)− ex3
+ 1
x6(f) lim
x→0
x2 sin(x2) + ex4 − 1
ln(1 + x4)
(g) limx→0
cos(2x2)− ex4
x sin(x3)(h) lim
x→0
sin(x8) + cos(3x4)− 1
ex8 − 1
F" =%(-(9$ 23&($2 )5'3&(/,2 $G*5 23&($2 .$ ;*%1)/(,2 ;,&, $)/*)%&,& *2 +,-*&$2 &$,(2 .$ k
:5$ %*&),' +H-(.,2 /,., 5', .,2 (05,-.,.$2 ,I,(J*"
(a)∞∑
n=0
enk = 9 (b) limx→0
e−x4 − cos(x2)
x4= k
K" #$2$)+*-+$& $' 23&($ .$ >,/-,5&() ,2 2$05()%$2 45)67$2D
(a) f(x) =1
1− x(b) f(x) =
1√1 + x
(c) f(x) =1
1 + x2
(d) f(x) =1√
1− x2(e) f(x) =
∫
sin x
xdx (f) f(x) =
∫
e−x2
dx
(g) f(x) =
∫
ln(1 + x)
xdx (h) f(x) = ln
(
1 + x
1− x
)
(i) f(x) = arcsin x
(j) f(x) = arccosx (k) f(x) = arctanx (l) f(x) = 3√1 + x
L" M,-/5-$ , ()%$0&,-
∫ t
0
13√1 + x4
dx 5%(-(9,).* $J;,)2<* $' 23&($ .$ ;*%1)/(,28 /$)%&,.,
$' 9$&*" #$%$&'()$ * %$&'* 0$&,- .$2%, $J;,)2<* *5 4,6, * 2$5 .$2$)+*-+('$)%* /*'
;$-* '$)*2 A %$&'*2 )<* )5-*2"
!NE
!"# $%&'(&)*&
! !
(a) 14
(b) 0 (c) 0 (d) 0 (e) @ (f) 0 (g) @ (h) @
(i) @ (j) π2
(k) e−2 (l) 0 (m) 0 (n) @ (o) 1 (p) 0
"! (a) un = 2n−1
3n(b) un = (−1)n−12n−1
3n(c) un = 2n−1
2n(d) un = n−1
n2
#! !
(a) decrescente (b) decrescente (c) decrescente (d) decrescente(e) decrescente (f) crescente (g) decrescente (h) nao$decrescente
%! & '()*+,-./ -0,1(23(4 50.' 6 *7/ '()*+,-./ 70,890,/ :.7.9/;/! <(* :.7.9( L 6 9/: )*(
1 ≤ L ≤ 5.
=! <( / '()*+,-./ >02 70,890,/ -2('-(,9(4 '(2? -0,1(23(,9(4 -07 :.7.9( L ≤ 5. @02674 '(
/ '()*+,-./ >02 70,890,/ ;(-2('-(,9( ,/;/ 50;(70' /A27/2!
B! C.-/ 5/2/ 0 .9(7 DEFG H09( )*( '( L = limn→+∞
un (,9I0 limn→+∞
un+1 = L. J07 .''04
/5:.-/$'( :.7.9(' (7 /7E0' :/;0' ;/ 2(:/KI0 ;( 2(-022+,-./ ;/;/ ( 0E967$'( )*( L =12
(
L+ kL
)
. &302/ E/'9/ .'0:/2 L.
L! C.-/ 5/2/ 0 .9(7 D-FG H09( )*( '( τ = limn→+∞
xn = limn→+∞
un+1
un
(,9I0 limn→+∞
un−1
un
=1
τ.
J07 .''04 /5:.-/$'( :.7.9(' (7 /7E0' :/;0' ;/ 2(:/KI0 ;( 2(-022+,-./ ;/;/ ( 0E967$'(
)*( τ = 1 +1
τ. &302/ E/'9/ .'0:/2 τ.
M! !
(a) Sk =k
2k + 1! J0,1(23( 5/2/
1
2(b) Sk =
8k
4k + 1! J0,1(23( 5/2/ 2
(c) Sk =k (k + 2)
(k + 1)2! J0,1(23( 5/2/ 1 (d) Sk = − ln(k + 1). C.1(23(
(e) Sk =1
3− 2k
3.5k! J0,1(23( 5/2/
1
3(f) Sk = 1− 1√
k + 1!J0,1(23( 5/2/ 1
(g) Sk =1
2− 1
(k + 2)!! J0,1(23( 5/2/
1
2(h) Sk =
5
2− 2
k + 1− 1
k + 2! J0,1(23( 5/2/
5
2
N! !
(a) F (b) F (c) F (d) F (e) V (f) V (g) F (h) F(i) F (j) F (k) V (l) V (m) V (n) V (o) V (p) F
O! Sk = 2− 2
2k + 1. & '62.( -0,1(23( 5/2/ 2.
! (a) S =1
4(b) S =
1
7(c) S =
7
24(d) & '62.( ;.1(23(
"! P(3(,;/G J D-0,1(23(,9(F4 C D;.1(23(,9(F4 Q D.,-0,-:*'.10FG
(a) C (b) C (c) C (d) D (e) D (f) C (g) C (h) C (i) C(j) D (k) C (l) C (m) D (n) D (o) C (p) D (q) C (r) C
" L
!" #$%$&'() * +,-&.$/%$&0$12 3 +'4.$/%$&0$12 5 +4&,-&,6784.-1)
(a) C (b) D (c) C (d) I (e) D (f) C (g) I (h) C (i) I (j) C (k) D (l) D (m) C
9" #$%$&'() * +,-&.$/%$&0$12 3 +'4.$/%$&0$12 5 +4&,-&,6784.-1)
(a) C (b) C (c) C (d) C
:" #$%$&'() * +,-&.$/%$&0$12 3 +'4.$/%$&0$12 5 +4&,-&,6784.-1)
(a) C (b) D (c) D (d) D (e) C (f) C (g) C (h) C (i) D (j) C (k) C
;" "
(a) (<8-670(=$&0$ (b) (<8-670(=$&0$ (c) (<8-670(=$&0$
(d) (<8-670(=$&0$ (e) '4.$/%$&0$ (f) (<8-670(=$&0$
(g) (<8-670(=$&0$ (h) ,-&'4,4-&(6=$&0$ (i) '4.$/%$&0$(j) ,-&'4,4-&(6=$&0$ (k) '4.$/%$&0$ (l) (<8-670(=$&0$
(m) ,-&'4,4-&(6=$&0$ (n) (<8-670(=$&0$ (o) ,-&'4,4-&(6=$&0$
>" "
(a) (<8-670(=$&0$ (b) ,-&'4,4-&(6=$&0$ (c) ,-&'4,4-&(6=$&0$
(d) (<8-670(=$&0$ (e) (<8-670(=$&0$ (f) (<8-670(=$&0$
(g) '4.$/%$&0$ (h) (<8-670(=$&0$ (i) '4.$/%$&0$
?" I @ - 4&0$/.(6- '$ ,-&.$/%A&,4( $ R @ - /(4- '$ ,-&.$/%A&,4(
(a) R = 1, I = [−1, 1) (b) R = 1, I = [−1, 1] (c) R =∞, I = (−∞,∞)(d) R = 1
4, I = (−1
4, 14) (e) R = 1
2, I = (−1
2, 12] (f) R = 4, I = (−4, 4]
(g) R = 3, I = (−5, 1) (h) R = 1, I = (3, 5) (i) R = 2, I = (−4, 0](j) R = 0, I = {1
2} (k) R = 3, I = [−3, 3] (l) R = 1
4, I = [1, 3
2]
(m) I = [4, 6), R = 1 (n) I = (−4, 0), R = 2 (o) I = (1− e, 1 + e), R = e
(p) I = [−32,−1
2], R = 1
2(q) I = [0, 2], R = 1 (r) I = (−3
2, 32), R = 3
2
B" [−1, 1], [−1, 1] $ (−1, 1), /$8C$,04.(=$&0$"
DE" "
(a)1
(1− x)2(b)
x
(1− x)2(c) 2 (d)
2x2
(1− x)3
(e) 4 (f) 6 (g) − ln(1 + x) (h) 2 ln 32
D " "
(a) f(x) =∞∑
n=0
(−1)nx3n (b) f(x) =∞∑
n=0
(−1)nx3n
4n+1
(c)f(x) =∞∑
n=0
(−1)n4nx2n+1
9n+1(d) f(x) =
∞∑
n=1
2n−1nxn+1
(e) f(x) =∞∑
n=1
nxn+2
2n+1(f) f(x) = −
∞∑
n=0
xn+1
(n+ 1)5n+1
(g) f(x) =∞∑
n=0
(−1)nx2n+3
n+ 1
DD" "
(a)∞∑
n=0
x8n+2
8n+ 2+K (b) −
∞∑
n=1
x2n−1
n(2n− 1)+K (c)
∞∑
n=1
(−1)n+1x2n−1
4n2 − 1+K
(d)∞∑
n=0
(−1)nx4n+3
(4n+ 3)(2n+ 1)+K
D ?
!" #$%&' ()*+,- ./- arctanx =∞∑
n=0
(−1)nx2n+1
2n+ 1- 0-1)$* 2&3& x =
√3
3.
4" #$%&' 0-,$5- +-,6) & +-,6)7 0-*8)./- ) 9:0$%- 0) *)6&+;,$) - */<*+$+/& :& -./&3=)
0&0&"
>" #$%&' 0-,$5- +-,6) & +-,6)7 0-*8)./- ) 9:0$%- 0) *)6&+;,$) - */<*+$+/& :& -./&3=)
0&0&"
?" (a)
√2
2(b)
√3
2(c) e3 − 1 (d) e
3
5
@" A:+-,5&8) 0- %):5-,BC:%$&'
−12≤ x ≤ 9
2- ,&$) 0- %):5-,BC:%$& R =
5
2.
D" A:+-,5&8) 0- %):5-,BC:%$&'
−23≤ x < 4.
E" #$%&' F)+- ./- & *G,$- 0&0& G B-)6G+,$%&H
!I"
∞∑
n=0
(−1)n(4n+ 4)(x− 1)n, $:+-,5&8) 0- %):5-,BC:%$&' 0 < x < 2.
!J" cosh(x3) =∞∑
n=0
x6n
(2n)!, ./- %):5-,B- 1&,& +)0) x ∈ R
! " #-*-:5)85$6-:+) -6 *G,$-* 0- (&%K&/,$: : f(x) =∞∑
n=1
x2n−1
n!./- %):5-,B- 1&,& +)0)
x ∈ R, )/ *-L&7 ) ,&$) 0- %):5-,BC:%$& G $:M:$+)"
!!" N&*+& $:+-B,&, +-,6) & +-,6)"
!4" f(x) =∞∑
n=0
(−1)n4n+1x2n+5
(n+ 1)(2n+ 5)%):5-,B- 1&,&
−12≤ x ≤ 1
2.
!>" #-*-:5)85$6-:+) -6 *G,$-* 0- (&%8&/,$:
(a)∞∑
n=0
(−1)n22n+1x2n+2
(2n+ 2)!(b)
∞∑
n=0
(−1)n22n+1x2n+3
(2n+ 1)!(c)
∞∑
n=0
3nxn
n!
(d)∞∑
n=0
(−1)nx2n
n!(e)
∞∑
n=0
(−1)n22nx2n
(2n)!(f)
∞∑
n=0
(−1)nx10n+2
(2n+ 1)!
(g)∞∑
n=1
(−1)nx2n−2
(2n)!(h)
∞∑
n=0
x2n+3
n!
!?" (a)2
3(b) − 2
3(c) 2 (d) − 5 (e) − 1 (f) 2 (g) − 3 (h) − 7
2
!@" (a) k = ln8
9(b) k = −1
2
JE
!" #$%$&'()'*+$&,( $+ -./*$% 0$ 123425/*&
(a)∞∑
n=0
xn (b) 1 +∞∑
n=1
(−1)n1.3.5. · · · .(2n− 1)xn
2nn!
(c)∞∑
n=0
(−1)nx2n (d) 1 +∞∑
n=1
1.3.5. · · · .(2n− 1)x2n
2nn!
(e)∞∑
n=0
(−1)nx2n+1
(2n+ 1)!(2n+ 1)+ C (f)
∞∑
n=0
(−1)nx2n+1
(2n+ 1)!+ C
(g)∞∑
n=0
(−1)nxn+1
(n+ 1)2+ C (h)
∞∑
n=0
2x2n+1
2n+ 1
(i) x+∞∑
n=1
1.3.5. · · · .(2n− 1)x2n+1
(2n+ 1)2nn!(j) − x−
∞∑
n=1
1.3.5. · · · .(2n− 1)x2n+1
(2n+ 1)2nn!
(k)∞∑
n=0
(−1)nx2n+1
2n+ 1(l) 1 +
1
3x+
∞∑
n=2
(−1)n2.5.8. · · · .(3n− 4)xn
3nn!
6"
∫ t
0
13√1 + x4
dx = t+∞∑
n=1
(−1)n1.4.7.10. · · · .(3n− 2)t4n+1
(4n+ 1).3nn!
778
![Schwarz, Roberto - Sequencias Brasileiras [Livro Completo]](https://img.pdfslide.tips/doc/110x75/5572011b4979599169a0cc6c/schwarz-roberto-sequencias-brasileiras-livro-completo.jpg)
