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GII HN DY S

http://e-learning.hcmut.edu.vn/

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DY S THC

Dy s l tp hp cc s c nh ch st nh n ln trong tp hp s t nhin N.

VD: 1/ xn = n2, n = 0, 1, 2,

2/ xn= 1/n, n = 1, 2,

3/ {xn} l cp s cng: a, a+d, a+2d,

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Cc cch cho dy s

21, /n nx n x n

1/Dng lit k:

VD: dy 1, 2, 3,; dy 1, 1/2, 1/3,

2/Dng tng minh:

{xn} cho dng biu thc gii tch ca bin n.

3/Dng quy np:

S hng i sau tnh theo cc s hng i trc

VD:

VD: dy 21 11 1, n n nx x x x

dy 11 2 11 1

2

, , n nnx x

x x x

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Dy n iu

{xn} l dy tngxn xn+1, vi mi n ln

{xn} l dy gimxn xn+1, vi mi n ln

Dy tng v dy gim gi chung l dy n iu.

B du = trong nh ngha ta gi l tng(gim) ngt.

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1.Xt hiu s: xn+1 xn(so vi 0)

2.Xt thng s: xn+1/xn(so vi 1)(dng cho dy s dng)

3.Xt o hm ca hm s f(x), vi f(n) = xn

Phng php kho st dy n iu:

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V d

1 11 2

/ :na x n

1 1 / 1 1 :

2nb x

n

gim1 11

1

n

n

x

x n

1

1

1

n nx x

n

0 tng

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2 3/ :

3 4n

nc x

n

Biu thc ging hm s, xt o hm

22 3 1( ) , ( ) 03 4 (3 4)

xf x f x x x

f(x) tng {xn} tng.

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Dy b chn

{xn} l dy b chn trn M : xn M, n N0

{xn} l dy b chn di m : xn m, n N0

{xn} b chn {xn} b chn trn v b chn di

VD: Xet tnh b chan cua cac day

/ 1 nc n / 3nb

2

1/a

n

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1 2 3 4 5 6 7 8, , , , , , , , , ,n nx x x x x x x x x x

DY CON

Cho {xn}, chn ra cc s hng t dy ny

1cch ty theoth t ch s tng dn ta

c 1 dy con ca {xn}.

VD:

{x2n 1}{x2n}

{x2n-1} = {x1, x3, x5, }

{x2n} = {x2, x4, x6, }

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GiI HN DY S

nh ngha n gin: {xn} c gii hn l a

khi n ra tc l xna khi n ln

Dy hit

0 00, : , nN N x a n N

0 00, : nN a x a n N

: limhu hann

n

a x

a

0Nx

0( )

nx n N

a a

nh ngha cht ch:

1x

2x

3x

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V d

lim 11n

n

n 1

1

1 1

n

nx a

n n

1 11 1

1nx n

n

Chng minh

0

1 11 1nn N n n x

Chn N0 1/, vi > 0 ( b)

* Vi = 10-3, tm N0?

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Tnh cht dy hi t

Dy hi t th b chn.an 0 v an a th a 0

an a v a < c th an

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Cc php ton trn dy hi t

lim , lim

lim lim lim

lim lim lim lim 0

lim lim 0 & lim 0n

K :

K :

n nn n

n n n n n n n

n n n n n n n n n

n n n

n n n

x y

x y x y

x y x y y

x x x x

lim tng(hiu, tch, thng, cn,)

=tng(hiu) lim

(hu hn)

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S HOI TU VA DAY CON

VD: dy {xn} = {(1)n} phn k

2

2 1

1 1

1 1V 2 day conn

n

x

x

lim xn

= a Mi dy con caxn

u a

Dy xnphn k 1 day con phan ky

2 day con co lim nhau

2

2 1

nn

n

x ax

x a

aH qu:

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GII HAN KEPCho 3 dy xn, yn, zn

0

lim lim

n n n

n nn n

x y z n N

x z a

lim

nn

y a

n n nx y z

a

Hqu: 0 & lim 0 lim 0n n n n n n

x y n y x

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Dy phn k ra v cng

Giihn = : khng th xt | xn a | !

Dy khng hi t gi l dy phn k:

Khng c giihn

Phn k ra vcng

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0 0lim , : ,> 0n nn

x M N N x M n N

0 0lim , : ,> 0n nn

x M N N x M n N

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V d

lim 2n

n

22 logn M n M

Chng minh

Vi M > 0 (ln) ty ,

Chn N0 > log2M + 1, ta c :

0 2log 2nn N n M M

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Cc php ton trn dy phn k ra

0

lim 0

0( 0),

nn

n

a

a n N

1lim 0n

na limn

n a 1/ Nu th

2/ Nu th limnn

a

()

lim nn

a

,lim nn

b c

lim nn

a

lim ( )

lim ,

n nn

n nn

a b

a b

neu c 0

, lim nn

b

lim ( )n nn

a b

lim n nn

a b

lim nn

a

, lim nn

b

3/

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GII HAN C BAN

3 / lim 1,n

nn

lim 0, 1n

n

na

a

4 / lim 1, 0nn

a a

1 lim

1 1 lim 0

n

n

n

n

a a

a a

0 lim

0 lim 0

n

n

n

n

2/. Ham mu:

1/. Luy tha:

lim 0, 0!

n

n

aa

n

ln5 / lim 0, 0

p

nn

n

ln p nn n a

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2 / lim 1

nn

e n

2 / lim na n

1 21 / lim lim 0n n

b nn

/ lim 2

nn

c 1 / lim 02

n

nd

2

/ lim 03

n

n

nf

V d

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7 DNG V NH

0, 0 , ,

0

0 01 ,0 ,

i vi 4 php ton cng, tr, nhn, chia:

i vi dng m nbna

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2

sin2 / lim 1n

n n

n

1000

3 / lim

n

n n

!1 / lim nn

n

n ! 1 2 10 0nn nn n n n n

2 2

sin

0 01 1

n n n

n n

Vi n 2000: 1000 10 02

n n

n

V d tng hp

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Tng cp s nhn

1 1

2 1 1lim 1 lim

1 1

n qn

n n

qq q q

q q

1 10 0

0 0 0

1

lim lim 1 1

n qn

n n

u q uu u q u q q q

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1 1 14 / lim 1

2 4 2

nn

11 1 2 1lim 2

1 1 2 1 1 2

n

n

2 1 3 9lim ,

3 2 8 32

n

5/ S

0

1 3 3 2 8,

2 2 4 3 21q u S

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25 / lim 1n

n n

2 2

2

1lim

1n

n n

n n

2 2

1 1 1 1lim lim 0. 0

21 1 1n n nn n n

26 / lim 1n

n n n

27 / lim 1 2n

n n

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8 / lim 3 2n n n

n

3

( 0 )

2 3n n 3 2n

3 1

n

3

lim 3nn

x

3 2n n n

nx

28 / lim 3 2n n n

nn

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TIEU CHUAN WEIRSTRASS

Dy tng & bchn trn th hit,

Dy gim & bchndi th hit

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VD: 1/Chng minh tntigiihn sau:

2 2 21 1 1lim 1 ...

2 3 n n

2 2

1 1

1 ... :2nx n

21 1 1 1

21 1

nxn n n n n

1 21

0

( 1)

tangn n nx x x

n

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2/Chng minh tnti v tm giihn dy s:

0 13, 12

nn

xx x

1n nx x

TIEU CHUAN WEIRSTRASS

Dng quy np chng minh xn > 2 (b chn

di)

n iu:

1

21 1 2

2 2n

k

xx

1 02

nx

Gs xk> 2,

12

n nx x

{xn} gim v b chn di nn hi t

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lim nx LGi:

Khi 1lim nx L

Ta li c 1 12n

n

x

x

Qua gii hn khi n, ta c

1 22

LL L

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0

11

n knn

kk

C

n n

2

1 1 12 1 1!

n

kk

n n k

1

2

1 1 22 2 3

1 1 22

n

k

k

B chn:

2

1 1 12

!

n

kk

n n n k

k n

2

1

2 !

n

k k

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PHA DANG VO NH 1

: lim 1 .

na

n

aa en

1 1

1 / lim 1VD :

n

n n e

2 32

2 / lim4

n

n

n

n

2 3

4 4 22

4

2 1lim 1

4Bien oi

nn n

ne

n e

1 .Dang