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i
Mc lc
Li ni u iii
Cc k hiu v khi nim vii
Bi tp
1 S thc 3
1.1 Cn trn ng v cn di ng ca tp cc s thc. Linphn s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Mt s bt ng thc s cp . . . . . . . . . . . . . . . . . . 11
2 Dy s thc 19
2.1 Dy n iu . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Gii hn. Tnh cht ca dy hi t . . . . . . . . . . . . . . 30
2.3 nh l Toeplitz, nh l Stolz v ng dng . . . . . . . . . 37
2.4 im gii hn. Gii hn trn v gii hn di . . . . . . . . 42
2.5 Cc bi ton hn hp . . . . . . . . . . . . . . . . . . . . . . 48
3 Chui s thc 63
3.1 Tng ca chui . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.2 Chui dng . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.3 Du hiu tch phn . . . . . . . . . . . . . . . . . . . . . . . 90
3.4 Hi t tuyt i. nh l Leibniz . . . . . . . . . . . . . . . 93
3.5 Tiu chun Dirichlet v tiu chun Abel . . . . . . . . . . . . 99
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3.6 Tch Cauchy ca cc chui v hn . . . . . . . . . . . . . . 102
3.7 Sp xp li chui. Chui kp . . . . . . . . . . . . . . . . . . 104
3.8 Tch v hn . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Li gii
1 S thc 121
1.1 Cn trn ng v cn di ng ca tp cc s thc. Linphn s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
1.2 Mt s bt ng thc s cp . . . . . . . . . . . . . . . . . . 131
2 Dy s thc 145
2.1 Dy n iu . . . . . . . . . . . . . . . . . . . . . . . . . . 145
2.2 Gii hn. Tnh cht ca dy hi t . . . . . . . . . . . . . . 156
2.3 nh l Toeplitz, nh l Stolz v ng dng . . . . . . . . . . 173
2.4 im gii hn. Gii hn trn v gii hn di . . . . . . . . 181
2.5 Cc bi ton hn hp . . . . . . . . . . . . . . . . . . . . . . 199
3 Chui s thc 231
3.1 Tng ca chui . . . . . . . . . . . . . . . . . . . . . . . . . 231
3.2 Chui dng . . . . . . . . . . . . . . . . . . . . . . . . . . 253
3.3 Du hiu tch phn . . . . . . . . . . . . . . . . . . . . . . . 285
3.4 Hi t tuyt i. nh l Leibniz . . . . . . . . . . . . . . . 291
3.5 Tiu chun Dirichlet v tiu chun Abel . . . . . . . . . . . . 304
3.6 Tch Cauchy ca cc chui v hn . . . . . . . . . . . . . . 313
3.7 Sp xp li chui. Chui kp . . . . . . . . . . . . . . . . . . 321
3.8 Tch v hn . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
Ti liu tham kho 354
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Li ni u
Bn ang c trong tay tp I ca mt trong nhng sch bi tp gii tch(theo chng ti) hay nht th gii .
Trc y, hu ht nhng ngi lm ton ca Vit Nam thng s dnghai cun sch ni ting sau (bng ting Nga v c dch ra ting Vit):
1. Bi tp gii tch ton hc ca Demidovich (B. P. Demidovich;1969, Sbornik Zadach i Uprazhnenii po Matematicheskomu Analizu,Izdatelp1stvo Nauka, Moskva)
v
2. Gii tch ton hc, cc v d v bi tp ca Ljaszko, Bojachuk,Gai, Golovach (I. I. Lyashko, A. K. Boyachuk, YA. G. Gai, G. P.Golobach; 1975, Matematicheski Analiz v Primerakh i Zadachakh,Tom 1, 2, Izdatelp1stvo Vishaya Shkola).
ging dy hoc hc gii tch.
Cn ch rng, cun th nht ch c bi tp v p s. Cun th haicho li gii chi tit i vi phn ln bi tp ca cun th nht v mt sbi ton khc.
Ln ny chng ti chn cun sch (bng ting Ba Lan v c dchra ting Anh):
3. Bi tp gii tch. Tp I: S thc, Dy s v Chui s (W. J.Kaczkor, M. T. Nowak, Zadania z Analizy Matematycznej, Czesc Pier-wsza, Liczby Rzeczywiste, Ciagi i Szeregi Liczbowe, WydawnictwoUniversytetu Marii Curie - Sklodowskiej, Lublin, 1996),
4. Bi tp gii tch. Tp II: Lin tc v Vi phn (W. J. Kaczkor, M.T. Nowak, Zadania z Analizy Matematycznej, Czesc Druga, Funkcje
iii
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iv Li ni u
Jednej ZmiennejRachunek Rozniczowy, Wydawnictwo UniversytetuMarii Curie - Sklodowskiej, Lublin, 1998).
bin dch nhm cung cp thm mt ti liu tt gip bn c hc v dygii tch. Khi bin dch, chng ti tham kho bn ting Anh:
3*. W. J. Kaczkor, M. T. Nowak, Problems in Mathematical Analy-sis I, Real Numbers, Sequences and Series, AMS, 2000.
4*. W. J. Kaczkor, M. T. Nowak, Problems in Mathematical Analy-sis II, Continuity and Differentiation, AMS, 2001.
Sch ny c cc u im sau:
Cc bi tp c xp xp t d cho ti kh v c nhiu bi tp hay.
Li gii kh y v chi tit.
Kt hp c nhng tng hay gia ton hc s cp v ton hchin i. Nhiu bi tp c ly t cc tp ch ni ting nh, Ameri-can Mathematical Monthly (ting Anh), Mathematics Today (tingNga), Delta (ting Balan). V th, sch ny c th dng lm ti liucho cc hc sinh ph thng cc lp chuyn cng nh cho cc sinhvin i hc ngnh ton.
Cc kin thc c bn gii cc bi tp trong sch ny c th tm trong
5. Nguyn Duy Tin, Bi Ging Gii Tch, Tp I, NXB i Hc QucGia H Ni, 2000.
6. W. Rudin, Principles of Mathematical Analysis, McGraw -HilBook Company, New York, 1964.
Tuy vy, trc mi chng chng ti trnh by tm tt l thuyt gipbn c nh li cc kin thc c bn cn thit khi gii bi tp trong chngtng ng.
Tp I v II ca sch ch bn n hm s mt bin s (tr phn khnggian metric trong tp II). Kaczkor, Nowak chc s cn vit Bi Tp GiiTch cho hm nhiu bin v php tnh tch phn.
Chng ti ang bin dch tp II, sp ti s xut bn.
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Li ni u v
Chng ti rt bit n :
- Gio s Phm Xun Ym (Php) gi cho chng ti bn gc tingAnh tp I ca sch ny,
- Gio s Nguyn Hu Vit Hng (Vit Nam) gi cho chng ti bngc ting Anh tp II ca sch ny,
- Gio s Spencer Shaw (M) gi cho chng ti bn gc ting Anhcun sch ni ting ca W. Rudin (ni trn), xut bn ln th ba, 1976,
- TS Dng Tt Thng c v v to iu kin chng ti bin dchcun sch ny.
Chng ti chn thnh cm n tp th sinh vin Ton - L K5 H oTo C Nhn Khoa Hc Ti Nng, Trng HKHTN, HQGHN, ck bn tho v sa nhiu li ch bn ca bn nh my u tin.
Chng ti hy vng rng cun sch ny s c ng o bn c nnhn v gp nhiu kin qu bu v phn bin dch v trnh by. Rt mongnhn c s ch gio ca qu v bn c, nhng kin gp xin gi v:Chi on cn b, Khoa Ton C Tin hc, trng i hc Khoahc T nhin, i hc Quc gia H Ni, 334 Nguyn Tri, ThanhXun, H Ni.
Xin chn thnh cm n.
H Ni, Xun 2002.
Nhm bin dch
on Chi
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i
Cc k hiu v khi nim
R - tp cc s thc
R+ - tp cc s thc dng
Z - tp cc s nguyn
N - tp cc s nguyn dng hay cc s t nhin
Q - tp cc s hu t
(a, b) - khong m c hai u mt l a v b
[a, b] - on (khong ng) c hai u mt l a v b
[x] - phn nguyn ca s thc x
Vi x R, hm du ca x l
sgn x =
1 vi x > 0,
1 vi x < 0,0 vi x = 0.
Vi x N,
n! = 1 2 3 ... n,(2n)!! = 2 4 6 ... (2n 2) (2n),
(2n 1)!! = 1 3 5 ... (2n 3) (2n 1).
K hiu(
nk
)= n!
k!(nk)! , n, k N, n k, l h s ca khai trin nhthc Newton.
vii
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viii Cc k hiu v khi nim
Nu A R khc rng v b chn trn th ta k hiu sup A l cntrn ng ca n, nu n khng b chn trn th ta quy c rngsup A = +.
Nu A R khc rng v b chn di th ta k hiu inf A l cndi ng ca n, nu n khng b chn di th ta quy c rnginf A = .
Dy {an} cc s thc c gi l n iu tng (tng ng n iugim) nu an+1 an (tng ng nu an+1 an) vi mi n N. Lpcc dy n iu cha cc dy tng v gim.
S thc c c gi l im gii hn ca dy {an} nu tn ti mt dycon {ank} ca {an} hi t v c.
Cho S l tp cc im t ca dy {an}. Cn di ng v cn trnng ca dy , k hiu ln lt l lim
nan v lim
nan c xc nh
nh sau
limn
an =
+ nu {an} khng b chn trn, nu {an} b chn trn v S = ,sup S nu {an} b chn trn v S 6= ,
limn
an =
nu {an} khng b chn di,+ nu {an} b chn di v S = ,inf S nu {an} b chn di v S 6= ,
Tch v hn
n=1
an hi t nu tn ti n0 N sao cho an 6= 0 vi
n n0 v dy {an0an0+1 ... an0+n} hi t khi n ti mt giihn P0 6= 0. S P = an0an0+1 ... an0+n P0 c gi l gi tr catch v hn.
Trong phn ln cc sch ton nc ta t trc n nay, cc hmtang v ctang cng nh cc hm ngc ca chng c k hiul tg x, cotg x, arctg x, arccotg x theo cch k hiu ca cc sch cngun gc t Php v Nga, tuy nhin trong cc sch ton ca Mv phn ln cc nc chu u, chng c k hiu tng t ltan x, cotx, arctan x, arccotx. Trong cun sch ny chng ti ss dng nhng k hiu ny bn c lm quen vi nhng k hiu c chun ho trn th gii.
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Bi tp
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Chng 1
S thc
Tm tt l thuyt
Cho A l tp con khng rng ca tp cc s thc R = (,).S thc x R c gi l mt cn trn ca A nu
a 6 x,x A.
Tp A c gi l b chn trn nu A c t nht mt cn trn.
S thc x R c gi l mt cn di ca A nu
a x,a A.
Tp A c gi l b chn di nu A c t nht mt cn di.
Tp A c gi l b chn nu A va b chn trn v va b chn di.R rng A b chn khi v ch khi tn ti x > 0 sao cho
|a| 6 x,a A.
Cho A l tp con khng rng ca tp cc s thc R = (,).S thc x R c gi l gi tr ln nht ca A nu
x A, a 6 x,a A.
Khi , ta vitx = max{a : a A} = max a
aA.
3
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4 Chng 1. S thc
S thc x R c gi l gi tr b nht ca A nu
x A, a x,a A.
Khi , ta vitx = min{a : a A} = min a
aA.
Cho A l tp con khng rng ca tp cc s thc R = (,). Gis A b chn trn.
S thc x R c gi l cn trn ng ca A, nu x l mt cntrn ca A v l cn trn b nht trong tp cc cn trn ca A. Tc l,
a 6 x,a A,
> o,a A, a > x .
Khi , ta vitx = sup{a : a A} = sup a
aA.
Cho A l tp con khng rng ca tp cc s thc R = (,). Gis A b chn di.
S thc x R c gi l cn di ng ca A, nu x l mt cndi ca A v l cn trn ln nht trong tp cc cn di ca A. Tc l,
a x,a A,
> o,a A, a < x + .
Khi , ta vitx = inf{a : a A} = inf a
aA.
Tin v cn trn ng ni rng nu A l tp con khng rng,b chn trn ca tp cc s thc, th A c cn trn ng (duy nht).
