giai tich 3

Gio trnh GII TCH 3

PGS. TS. Nguyn Xun Tho Email: [email protected]

PHNG TRNH VI PHN V L THUYT CHUI BI 1. CHNG I. L THUYT CHUI

1. i cng v chui s nh ngha iu kin cn chui hi t

Cc tnh cht c bn

t vn : 1 1 1 11 22 4 8 2n

+ + + + + + =

C phi l c cng mi cc s hng ca v tri th thnh v phi? 1 + ( 1)+1 + ( 1) + .... = ? 1. Chui s: nh ngha: Vi mi s t nhin n, cho tng ng vi mt s thc an, ta c dy s k hiu l { }na . nh ngha:

Cho dy s {an}, ta gi tng v hn 1 2 3a a a+ + + l chui s, k hiu l 1

nn

a

=

,

an l s hng tng qut. Sn = a1 + a2 + a3 + ... + an l tng ring th n. Nu lim n

nS S

= th ta bo chui hi t,

c tng S v vit: 1

nn

a S

=

= .

Khi dy {Sn} phn k th ta bo chui 1

nn

a

=

phn k.

V d 1. Xt s hi t v tnh 0

n

n

q

=

1

2 11 , 11

nn

nqS q q q q

q

+

= + + + + = + + + + = +

Do Sn c th ln bao nhiu tu , nn c lim nn

S

=

Chui cho phn k

V d 4. Chui nghch o bnh phng: 21

1

n n

=

( )2 2 21 1 1 1 1 1 1 1 11 1 1

2.2 3.3 . 1.2 2.3 12 3nS

n n n nn= + + + + = + + + + < + + + +

1 1 1 1 1 1 1 1 11 2 21 2 2 3 3 4 1n n n

= + + + + + =


1 1n

n

n

=+ phn k

V d 6. ( ) ( ) ( )1

1 1 1 1 1n

n

=

= + + + +

C ( ) chn l.

1lim 11

n

n

n

n

=

Khng tn ti ( )lim 1 nn

( )1

1 n

n

=

phn k.

V d 7. Tm tng (nu c) ca chui s sau ( )223 5 2 14 36 1

n

n n

++ + + +

+ (S: 1)

V d 8. 1

11

n

n

n

n

=

+ (PK)

Tnh cht. Gi s lim , limn nn n

a a b b

= =

( )lim n nn

a b a b

+ = + ( )lim .n n

na b a b

=

lim , 0.nn n

a a bb b

=

2. Chui s dng nh ngha Cc nh l so snh

Cc tiu chun hi t

1. nh ngha: 1

, 0n nn

a a

=

>

Nhn xt. 1

nn

a

=

hi t khi v ch khi Sn b chn.

Trong bi ny ta gi thit ch xt cc chui s dng 2. Cc nh l so snh. nh l 1. Cho hai chui s dng, n na b , n tu hoc t mt lc no tr i

1n

n

b

=

hi t 1

nn

a

=

hi t

1n

n

a

=

phn k 1

nn

b

=

phn k

PGS. TS. Nguyn Xun Tho Email: [email protected] Chng minh.

1 2 1 20

n n

n n

a a a b b bS T

+ + + < + + +

<

Rt ra cc khng nh.

V d 1. 1

13 1nn

=+

Chui dng 3 1 3

1 13 1 3

n n

n n

+ >

, n tu , chn m sao cho 2mn < , c

( ) ( )

( ) ( ) ( )

2 1 1

1

1 2 11 1 1

1

1 1 1 1 1 112 3 4 7 2 2 1

2 4 2 1 1 11 12 4 22 2 2

1 1 1, 0 1

1 1 2

mn p p p p p pm m

m

p p p p mm p p

m

p

S S

aa

a a

= + + + + + + + + +

+ + + + = + + + +

= < < = <

Dy Sn b chn trn 1

1p

n n

=

hi t.

KL: Chui hi t vi p > 1 v phn k vi 0 < p 1.

V d 6. 3

1

1

3n n

= +

Chui dng

3 3 / 23

1 133 1

nan n

n

= =

+ +

; 3 / 21

nbn

=

lim 1nn n

a

b=

1n

n

b

=

hi t

31

1

3n n

= + hi t

PGS. TS. Nguyn Xun Tho Email: [email protected] V d 7

a1) ( )2ln 1 2 1

n

n n

=

+ + (PK) a2) ( )2sin 1 1

n

n n

=

+ (PK)

b1) 21

sin2n

nn

=

pi

(PK); b2) ( )11

1 2 1nn

n

=

(HT)

c1) 5

1

cos

1n

n n

n

=

+

+ (HT) c2) 3

1

sin1n

n n

n

=

+

+ (PK)

d1) ( )2

2 1n

n n

=

+ (PK) d3) ( )12

1nn

n e

=

(PK)

d3)3 7 3

1

1sin

2 3n

n

n n

=

+

+ + (HT)

e) Xt s hi t 1)

=

4 51

ln

n

n

n (HT) 2)

+

11

arcsin lnnn

(PK)

3) pi

=

+

2 3

1ln 1 arctan

2nn

n (HT)

3) Cc tiu chun hi t a) Tiu chun DAlembert

1lim nn n

a la

+

=

Khi 1l < 1

nn

a

=

hi t

Khi 1l > 1

nn

a

=

phn k.

Chng minh l < 1: T 1lim n

n n

a la

+

= , chn > 0 b l + < 1 1n

n

a

a+

< l + , n n0.

Mt khc c 00

0

11

1 2. .

nn nn n

n n n

aa aa a

a a a+

= ( ) 00

n nnl a

+ 0, n

Do lim nn

a l

=


l > 1: T 1lim nn n

a la

+

= , chn b l > 1 1 1n

n

a la

+ > > an + 1 > an

phn k Nhn xt. Khi l = 1 khng c kt lun g

V d 1. 1

1!

nn

=

1 0!n

an

= >

( ) ( )1 1 1 ! 1lim lim : lim lim 0 1

1 ! ! 1 ! 1n

n n n nn

a n

a n n n n+

= = = =

( )1

1 3 3 3:1 ! ! 1

n nn

n

a

a n n n

++

= =

+ +

1lim 0 1nn n

a

a+

= <

Chui cho hi t

V d 3. Xt s hi t, phn k ca chui ( )( )1.3.5 2 11 1.3 1.3.5

2 2.5 2.5.8 2.5.8 3 1n

n

+ + + +

( )( )

1.3.5 2 1 02.5.8 3 1n

na

n

= >

( ) ( )( )( )

( )( )

1

1

1.3.5 2 1 2 1 1.3.5 2 1 2 1:

2.5.8 3 1 3 2 2.5.8 3 1 3 22lim 13

n

n

n

n n

n n na n

a n n n n

a

a

+

+

+ += =

+ +

= 1

nn

a

=

phn k

Nhn xt. Nu l = 1, khng c kt lun g

V d 5. 1

2 13 2

n

n

n

n

=

+

2 1 03 2n

na

n

= > +

2 13 2

nn

na

n

=

+

2lim 13

nn

na

= <

Chui cho hi t

V d 6. Xt s hi t, phn k 2

1

1 n

n

n

n

=

+ (PK)

V d 7.

a1) 2 ln2

21

3 14 cos

n n

n

n n

n n

=

+ +

+ (HT) a2)

=

+ +

+

3 ln2

21

2 13 sin

n n

n

n n

n n (HT)


a3)( )

2

21

5

2 1

n n

nnn

n

n

= + (HT)

b1) ( )4

1

23

n n

n

n

n

+

=

+

+ (HT) b2) ( )4

1

32

n n

n

n

n

+

=

+ + (PK)

c) ( )

= +

2

21

5

3 1

n n

nnn

n

n (HT)

c) Tiu chun tch phn C mi lin h hay khng gia:

( ) lim ( )b

ba a

f x dx f x dx

+=

v 1 1

limk

n nkn n

a a

= =

=

1 2 11 1

( ) ( )n n

nf x dx a a a a f x dx + + + + , + =lim ( ) 0x f x

Nu f(x) l hm dng gim vi mi x 1, f(n) = an, khi

1n

n

a

=

v 1

( )f x dx

cng hi t hoc cng phn k.

V d 8. 2

1ln

nn n

=

1( )ln

f xx x

= dng, gim vi 2x v c +

=lim ( ) 0x

f x

( ) ( ) ( ) ( )( )22 2

ln( ) lim lim ln ln lim ln ln ln ln2ln

bb

b b n

d xf x dx x bx

= = = =

1

( )f x dx+

phn k

2

1ln

nn n

=

phn k

Tng qut c th xt ( )21

ln pn n n

=

hi t ch khi p > 1.

Hnh 14.4


V d 9. Chng minh rng: 1 1 11 ln22 3 4

+ + =

[ ] [ ]

21 1 1 1 1 1 1 1 1 11 12 3 4 2 1 2 3 2 1 2 4 2

1 1 1 1 1 1 1 1 1 1 1 11 2 1 12 3 2 2 4 2 2 3 2 2 3

1 1ln2 (1) ln (1) , lim 1 ln2

n

n

Sn n n n

n n n n

n o n o nn

= + + + = + + + + + +

= + + + + + + + = + + + + + + + +

= + + + + = + + +

=

vi

ln2 (1) ln2o n+ khi Mt khc ta c

( )

2 1 2

2 1 2

1

1

12 1

lim lim ln2

1ln2

n n

n nn

n

n

S Sn

S S

n

+

+

+

=

= ++

= =

=

V d 10. Tng t nhn c 1 1 1 1 1 31 ln2.3 2 5 7 4 2

+ + + + =

V d 11. Xt s hi t phn k ca chui s sau

a) ( )21

1ln

2nn

n

=+

(HT); b) ( )

( )21ln 1

3n

n

n

=

+

+ (HT) c) 2

2

ln3n

n

n

=

(HT)

Happy new year 2011 !Happy new year 2011 !Happy new year 2011 !Happy new year 2011 !

PGS. TS. Nguyn Xun Tho [email protected]

HAPPY NEW YEAR 2011HAPPY NEW YEAR 2011HAPPY NEW YEAR 2011HAPPY NEW YEAR 2011

PHNG TRNH VI PHN V L THUYT CHUI BI 2

3. Chui s vi s hng c du bt k Chui vi s hng c du bt k Chui an du

Tnh cht ca chui hi t tuyt i

1. t vn . 2. Chui vi s hng c du bt k

nh ngha:

=

1

nn

a c gi l hi t tuyt i

=

1

nn

a hi t. Chui

=

1

nn

a c gi

l bn hi t

=

1

nn

a phn k v

=

1

nn

a hi t.

nh l.

=

1

nn

a hi t 1

nn

a

=

hi t.

V d 1. Xt s hi t tuyt i ca chui s sau

a) ( ) +

=

2

21

12

n n

nn

n ; b)

=

21sin

n

n

c) ( )( )pi=

+1sin 2 3

n

n

(HTT) d)

=

31

sin

n

n

n (HTT)

Hng dn.

a) ( ) +

=

2

21

12

n n

nn

n

+) Xt

=

1 2

nn

n

+) +

=

PGS. TS. Nguyn Xun Tho [email protected] Nhn xt.

1/ Nu

=

1

nn

a phn k theo tiu chun DAlembert hoc Cauchy

=

1

nn

a phn k

2/

=

1

nn

a phn k

=

1

nn

a phn k (ng hay sai?)

3. Chui an du

nh ngha. ( )

=

> 11

1 , 0n n nn

a a c gi l chui an du

Ch . ( )

=

>1

1 , 0n n nn

a a cng c gi l chui an du.

nh l Leibnitz

Dy { }na gim, > 0na , lim 0nn

a

= ( )

=

11

1 n nn

a hi t v c ( ) 1 11

1 n nn

a a

=

Chng minh: +) = 2n m : C ( ) ( ) ( )

= + + + 2 1 2 3 4 2 1 2m m mS a a a a a a { }2mS tng ( ) ( ) ( )

=


i) ( )

=

+ 2

11

2 1n

n

n

n (PK)

k) ( )

=

+

+ 1112

nn

n

n

n (PK)

l) ( )

=

+ 1 2

1

11 lnn

n

n

n (HTT)

m) ( )

=

11

ln1 n

n

n

n (Bn HT)

o) ( ) ( )3 7 3

1

1 sin 2,

2 3n

n n

n n

=

+

+ + (HTT)

p) ( )

=

1

1ln

n

nn n

(Bn HT)

Hng dn.

b) +) ( )

=

1

1

1 n

n n l chui an du

+)

1n

gim v c

=

1lim 0n n

+) Hi t theo Leibnitz

+)

=

1

1

n n phn k bn hi t

d) +) ( )

=

1

1

16 5

n

n

n

n l chui an du

+)

=

1lim6 5 6n

n

n

=

1 6 5n

n

n phn k

+) ( )

1lim 16 5

n

n

n

n

+) ( )

=

1

16 5

n

n

n

n phn k.

4. Tnh cht ca chui hi t tuyt i

a)

=

=1

nn

a S chui s nhn c t chui ny bng cch i th t cc s hng

v nhm tu cc s hng cng hi t tuyt i v c tng S

b) Cho

=

=1

nn

a S ,

=

1

nn

a phn k c th thay i th t cc s hng ca n

chui thu c hi t v c tng l mt s bt k cho trc hoc tr nn phn k.

nh ngha. Cho

= =

1 1

,n nn n

a b , khi ta nh ngha php nhn chui:

= = =

=

1 1 1n n n

n n n

a b c , 11

n

n k n kk

c a b + =

=

c)

=

= 11

nn

a S ,

=

= 21

nn

b S

= =

= 1 2

1 1n n

n n

a b S S

V d 3.a) Xt s hi t ca tch cc chui s sau: 1

1

n n n

=

v 11

12nn

=

.


b) Xt s hi t ca chui s ( )

= =

+

+ 1 2

1 1

1 21 tan .ln1

nk

n k

n kn kk k

Hng dn.

a) +)

=

1

1

n n n hi t tuyt i

+)

=

11

12nn

hi t tuyt i

+)

= =

1

1 1

1 1.

2nn nn n hi t

HAVE A GOOD UNDERSTANDING!HAVE A GOOD UNDERSTANDING!HAVE A GOOD UNDERSTANDING!HAVE A GOOD UNDERSTANDING!


PHNG TRNH VI PHN V L THUYT CHUI BI 3

4. Chui hm s t vn . 1. Chui hm s hi t nh ngha: Cho dy hm s ( ){ }nu x xc nh trn X , ta nh ngha chui hm s

( ) ( ) ( )

=

+ + 1 21

nn

u x u x u x (1)

( )

=

1

nn

u x hi t ti 0x chui s ( )

=

01

nn

u x hi t

( )

=

1

nn

u x phn k ti 0x chui s ( )

=

01

nn

u x phn k

Tp cc im hi t ca (1) gi l tp hi t ca n. Tng ca chui hm s l hm s xc nh trong tp hi t ca n. V d 1. Tm tp hi t ca cc chui hm s sau

a)

=

11

n

n

x b)

=+

2 21

cos

n

nx

n x c)

=

1

1x

n n ( 1x > ) d)

=

1 !

n

n

x

n ( )

e) ( )( )

=

+

+

2

21

sin 2 43 1n

n x

n ( ) f) ( )

=

1 cos1

1 n n xn

e ( 2 22 2

k x kpi pipi pi + < < + )

g) ( )( )+

=

1

1

15 3

n

nnn n x

( 135

x > )

Hng dn.

a)

=

11

n

n

x

+) Xt chui s

=

101

n

n

x (2)

+) (2) hi t vi

PGS. TS. Nguyn Xun Tho [email protected] V d 2. Tm tp hi t ca cc chui hm s sau

a) 1) ( ) ( ) +

=

+

1 2 3

21

13 2 3

n n

nn

x

n ( 3 3x < )

2) ( )

=+ +

1

11 1 nn n x

( 0 2x x> )

3) ( )

=+ +

31

11 2 nn n x

( 1 3x x> )

b) 1) ( )

=

+

3

221

4 3

1

n

n

n x

xn ( 3 ;1

5

)

2) ( )

=

+

22

1 111

n n

n

x

xn ([ )0 ; + )

c) ( )( )

=

+

+ +

2

0

11 2

n

n

x x

n n (0 1x )

2. Chui hm s hi t u

nh ngha. ( )

=

1

nn

u x hi t u n ( )S x trn tp X > 0 b tu

( ) 0 n : ( ) > 0n n , ta c ( ) ( ) 0 b tu

( ) 0 n : ( ) > > 0p q n , ta c ( ) ( ) < ,p qS x S x x X .

Tiu chun Weierstrass. Nu c ( ) , ,n nu x a n x X v

=

1

nn

a hi t

( )

=

1

nn

u x hi t tuyt i v u trn X .

V d 3. Xt s hi t u ca chui hm ( )

=

+

1

2 21

1 n

n x n

+) ( )

+

1

2 2 21 1

,

n

xx n n

+)

=

21

1

n n hi t

+) Chui cho hi t tuyt i v u trn V d 4. Xt s hi t u ca chui hm

a)

=

+

2 21

sin,

n

nxx

n x (HT) b) [ ]

=

31

, 2 ; 22

n

nn

xx

n n (HT)

c)

=

1

cos,

3

nn

nxx (HT) d) ( ) ( )

=

21

11 , 1; 1

nn

n

xx

n (HT)

e)

=

+

5 21

,

1

n

nxx

n x (HT) f)

=

>1

, 0!

n

n

xx

n (HTK)

Hng dn.


b) +) 4 / 331

, 22

n

n

xx

n n n +)

=

4 / 31

1

n n hi t

+) Chui cho hi t u v hi t tuyt i trn [ ]2 ; 2 . V d 5. Xt s hi t u ca chui hm

a) 1)

=

+

1

21 0

sin ,1

n

n

xdxnx x

x (HT) 2)

=

+

1

21 0

cos ,1

n

n

xdxnx x

x (HT)

b) 1) [ ]

=

+ +

+ 11 2 1

, 1; 123

n

nn

n xx

x (HT)

2) [ ]

=

+ +

+ + 2

1

1 2 1, 1; 1

2 2

n n

n

n xx

n x (HT)

c) Chng minh rng chui hm

=

1

2x

nx

n

e hi t u vi 0x

d) 1) Chng minh rng chui ( )

=

+ + 2

0

11

n

n x n hi t u trn

2) Chng minh rng chui ( )

=

+ + 2

0

12

n

n x n hi t u trn

3. Tnh cht ca chui hm s hi t u

nh l 1. Chui ( )

=

1

nn

u x hi t u v ( )S x trn X , ( )nu x lin tc trn X , vi

n ( )S x lin tc trn X .

nh l 2. ( )

=

1

nn

u x hi t u n ( )S x trn [ ];a b , ( )nu x lin tc trn [ ];a b , n

( ) ( ) ( )

= =

= =

1 1

b b b

n nn na a a

S x dx u x dx u x dx

nh l 3. ( ) ( )

=

=1

nn

u x S x trn ( );a b , cc hm ( )nu x kh vi lin tc trn ( );a b ,

( )

=

1

nn

u x hi t u trn ( );a b ( )S x kh vi trn ( );a b v c

( ) ( ) ( )

= =

= =

1 1n n

n n

S x u x u x

PGS. TS. Nguyn Xun Tho [email protected] V d 6. Xt tnh kh vi ca cc hm sau

a) ( ) ( )

=

=

+11 n

n

xf xn x

; b) ( )

=

= 21arctan

n

xf xn

( ( )2

4 21

,

n

nf x xn x

=

= +

)

Hng dn. a) +) x n l chui an du hi t theo Leibnitz +) ( ) ( ) = + 2n

nu x

n x lin tc

=

1

, nn

x n u hi t u theo Dirichlet

+) ( ) ( ) ( )

=

= +

21

1 ,n

n

nf x x nn x

V d 7 a) Tm min hi t v tnh tng 1) ( ) ( )

+

=

+3 2

0

113 1

nn

n

x

n ((0 ; 2] ,

21 1 2 3( 1) ln arctan3 3 3 6 33 3

x xS xx x

pi = + +

+ )

2) ( ) ( )+

=

+

+3 2

0

113 1

nn

n

x

n (( 2 ; 0) ,

21 2 1 2 1( 1) ln arctan3 3 3 6 31

x xS xx x

pi+ + = + + +

+ + )

b) Tm min hi t v tnh tng 1) ( ) ( )

=

+1

1

1 1n

n

n

xn

; 2) ( ) ( )( )

=

+ 11

1 1 1n n

n

n x ((0 ; 2) , 2

21xS

x

= )

Hng dn. b1) Hi t vi +

PGS. TS. Nguyn Xun Tho [email protected] PHNG TRNH VI PHN V L THUYT CHUI

BI 4 5 Chui lu tha

nh ngha Cc tnh cht Khai trin thnh chui lu tha t vn 1. nh ngha. 20 1 2 nna a x a x a x+ + + + + (1)

K hiu l 0

nn

n

a x

=

, na l cc s thc, x l bin s.

Ta bo chui lu tha hi t (phn k) ti 0x chui s 00

nn

n

a x

=

hi t (phn k),

chui 0

nn

n

a x

=

hi t trn khong ( );a b chui s 00

nn

n

a x

=

hi t, 0x tu ( ; )a b .

V d 1. 20

1nn

x x x

=

= + + +

bit hi t khi 1x < , c 0

11

n

n

xx

=

=

Phn k khi 1x

nh l 1 (Abel). 0

nn

n

a x

=

hi t ti 0 0x hi t tuyt i ti 0:x x x<

Chng minh. +) 01

nn

n

a x

=

hi t 0lim 0nnn

a x

= 0 0,n

na x M n N

+) 0 00 0

n nn n

n nx x

a x a x Mx x

=

+) 0

1xx

< 01

n

n

xMx

=

hi t (nh l so snh 1) 0

nn

n

a x

=

hi t tuyt i

Nhn xt. T nh l Abel suy ra: Nu 0

nn

n

a x

=

phn k ti 0x phn k ti 0:x x x>

nh l 2. Nu 1lim nn n

a

a+

= (hoc lim n n

na

= ) th bn knh hi t R ca chui lu

tha 1

nn

n

a x

=

c xc nh bi 1

, 0

0,, 0

R

< <

= = + =

giai tich 3

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