Сборник задач по высшей математике 2 том

00

lim an =O?

1.1.132.

MO)KHO JIH YTBep)K,ll;aTb, qTO PM

Lan CXO,ll;HTCH, eCJIH

n=l

n---+oo

00

1.1.133.

.HBJIHeTCH JIH He06xo,ll;HMbIM MH CXO,ll;HMOCTH pH,ll;a

L an YCJIo-

BHe:

n=l

a)

lim an ::j:. 2j

n---+oo

6) He Bce qJIeHbI pH,ll;a -

qHCJIa an -

paBHbI 2j

B)

lim an ::j:. OJ

n---+oo

r)

He Bce qJIeHbI pH,ll;a -

qHCJIa an -

paBHbI O?

1.1.134.

BepHO JIH, qTO

a) eCJIH pH,ll; CXO,ll;HTCH, TO ero qaCTHqHble CYMMbI OrpaHHqeHblj

6) eCJIH qaCTHqHble CYMMbI pH,ll;a OrpaHHqeHbI, TO pH,ll; CXO,ll;HTCH?

1.1.135. Cym;ecTByeT JIH PM, KOTOPblii

a) no npH3HaKY ~aJIaM6epa CXOMTCH, a no npH3HaKY KomH -

pacXO,ll;HTCHj

6) no npH3HaKY KomH CXO,ll;HTCH, a no npH3HaKY ~aJIaM6epa -

paCXOMTCHj

B) no npH3HaKY ~aJIaM6epa pacXO,ll;HTCH, a no HHTerpaJIbHOMY

npH3HaKY -

CXOMTCH?

00

1.1.136.

'thoMO)KHO CKa3aTb 0 CXOMMOCTH pH,ll;a L (an + bn ), ecJIH

n=l

00

00

a)

pH,ll;bI

L

an H

L

bn CXO)l;HTCHj

n=l

n=l

6) PH)l;bI

00

00

L

an H

L

bn pacXO)l;HTCHj

n=l

n=l

00

00

B) pH,ll; L

an CXOMTCH, a pH,ll;

L

bn pacXO,ll;HTCH?

n=l

n=l

00

(an + bn) CXOMTCH, CJIe)l;yeT JIH, qTO

1.1.137.

113 Tom, qTO PM

L

n=l

00

00

a) 06a pH,ll;a

L

an H

L

bn CXO)l;HTCHj

n=l

n=l

6) 06a pH,ll;a

00

00

L

an H

L

bn pacXO)l;HTCHj

n=l

n=l

00

00

B)

O,ll;HH H3 PH,ll;OB L

an H L

bn CXOMTCH, a ,ll;pyroii -

pacxo-

n=l

n=l

MTCH?

v'n2 + n+ 1 - v'n2 -

n - 1

1.1.138.

IIccJIe,ll;oBaTb Ha CXOMMOCTb pH,ll;

00

L

n

.

n=l

enn'

1.1.139.

00

IIccJIe,ll;oBaTb Ha CXOMMOCTb pH,ll;

L

----n=-.

n=l

n

00

1.1.140.

IIccJle)];oBaTb Ha CXO)J,HMOCTb PM E an, r)];e

n=l

3k _ 1

n

= 2k -1;

{ 4k - 1 '

k = 1,2, ...

an =

3k -

1

n = 2k,

7'

a) no npH3HaKy,I1;anaM6epa;

6) no npH3HaKY KOWH.

00

00

1.1.141.

TIpHBecTH npHMep )];ByX PH)];OB

E an H

E bn , )];JlH KOTOPbIX PH)];

n=l

n=l

00

00

E (an + bn ) CXO)];HTCH, a PM

E (an -

bn ) paCXO)];HTCH.

n=l

.

nn

n=l

1.1.142.

,I1;oKaJKHTe, '"ITO

hm

-- 2 = 0, HCCJle)];OBaB Ha CXO)];HMOCTb PH)];

00 nn

n-too

(n!)

E -

n=l (n!)

2 •

.

(n!)n

1.1.143. BhI'"IHCJlHTenpe)];eJl:

hm --2-.

n-too

nn

00

al - a2 + as - a4 + ... + (-It+1an + ... = L)-It+ 1an ,

(2.1)

n=l

HJIH

00

-al + a2 - as + an + ... + (-lta n + ... = L(- It an,

(2.2)

n=l

I')];eBce an - nOJIO:JKHTeJIbHhIe )];eil:cTBHTeJIbHhIe 'IHCJIa(an> 0, n = 1,2, ... ).

$

1)

al > a2 > as > ... > an > ... (a6cOJIIOTHhIe BeJIH'IHHhI'lJIeHOBpH)];a

MOHOTOHHO y6hIBaIOT)j

2)

lim an = 0 (06IIIHiI: 'IJIeHpH)];a CTpeMHTCH K HyJIIO npH n --+ 00),

n ..... oo

KonepeMeHHhIM.

$

ex>

TeopeMa 1.4. nYCTb AaH 3HaKonepeMeHHbliii PJlA ~ an, rAe an -

npOM3BOnb-

n=l

ex>

Hble 'lMCna(AeiiicTBMTenbHble MnM KOMnneKcHble). EcnM PJlA

~ lanl. COCTaaneH-

n=l

00

Hbliii M3 a6conIOTHbix BenM'lMHero 'lneHOB,CXOAMTCJI, TO AaHHbliii PJlA

~ an TaK)Ke

CXOAMTCJI.

n=l

B 3TOM c.rry'lae3HaKOnepeMeHHbli!: pH,n;

ex>

~ an H8.3b1BaeTCH a6COJl70mtW CXOOSl-

UfUMCJI.

n=l

ex>

ex>

~ Ec.rrH 2Ke 3HaKOnepeMeHHbli!: pH,n;

~ an CXO,n;HTCH, a pH,n;

~ lanl

pacXO,n;HTCH,

00

n=l

n=l

TO ,n;aHHbli!: pH,n; ~ an Ha3bIBaeTCH YCJl06'HO CXOOSl~UMCJI.

~

n=l

ex>

ex>

,Il;.rrH OTBeTa Ha Bonpoc 06 a6cOJIIOTHoi!: CXO,n;HMOCTH pH,n;a

~ an K pH.n;y ~ lanl

n=l

n=l

ex>

ex>

113 pacXO,n;HMOCTH pH,n;a

~ Ian I pacxo,n;HMOCTb pH,n;a ~ an,

Boo6IIIe I'OBOPH,

n=l

00

n=l

He c.rre.n;yeT. O,n;HaKO, ec.rrH, npHMeHHH K

pH.n;y

~ lanl npH3HaK ,1l;aJIaM6epa (HJIH

n=l

npH3HaK KOillH), nOJIY'laeMnpe,n;M

lim

n

1

I= l

> 1 (HJIH

lim

VIan I = l

> 1),

Iaa+

n~oo

n

n~oo

ex>

ex>

TO B 3TOM c.rry'lae06a pH,n;a -

~ Ian I H

~ an - paCXO,ll;HTCH.

n=l

n=l

= bn + ien, r,n;e

IIycTb {an} - nOc.rre,n;OBaTeJIbHOCTb KOMnJIeKCHblX 'IHCMan

,n;ei!:cTBHTMbHble 'IHc.rra,Il;JIH JII060ro n = 1,2, ... PH,n;

ex>

bn H Cn -

~ an (T. e. pH,n;

n=l

00

~(bn + icn )) CXO,n;HTCH Tor,n;a H TOJIbKO Tor,n;a, Kor,n;a CXo,n;HTCH,Il;Ba pH,n;a -

~ bn

00

00

00

00

n=l

H ~ Cn , npH'IeMB 3TOM c.rry'lae ~ an

= ~ bn + i

~ cn .

n=l

n=l

n=l

n=l

1.2.1.

00

1

IIccJIe,D;oBaTb Ha CXO,D;HMOCTb pH,D; E

(_I)n ..(ii

n=l

2

n - l

00

<) 1. IIccJIe.n;yeM Ha CXO,lJ;HMOCTb pH,D; E an H3 a6COJIIOTHbIX BeJIH'IHH'IJIe-

~OB ,D;aHHOro pH,D;a:

n=l

00

00

~an = ~ 2Jn-l'

00

1

TaK KaK 2..(ii -1 < 2..(ii, TO

1

>

CpaBHHM 9TOT PM C PH,D;OM E

.r,;:'

..(ii

n=12yn

2

n - l

2yn

n=12yn

001

001

1

< 1). 3Ha'lHT,no I-MY npH-

E r,;:

(KaK PM ,nHpHXJIe E p

npH p = -2

n=l yn

n=l n

00

1

3HaKY cpaBHeHHH pacXO,D;HTCH H pH,D; E

.jii

n=12

n - l

an =

1

1 >

1

=an +l·

2.jii -

2v'n + 1 - 1

1·

1·

1

0

1man =

1m

=.

n-+oo

n-+oo 2.jii

+ 1

n03TOMY ,D;aHHbIA pH,D; CXO,D;HTCH YCJIOBHO.

•

00

(_I)n+!

1.2.2.

IIccJIe,D;oBaTb Ha CXO,D;HMOCTb PH)J;

E

2

1

n=l n -

nn

00

a 1. IIccJIe,D;yeM pH,D; E an H3 a6COJIIOTHbIX BeJIHqHH qJIeHOB ,D;aHHOrO pH,D;a:

n=l

00

00

1

1

1

1

~

~

+ 4 -

+ ...

~ an = ~

2n - In n = "2

In 2 + 6 - In 3

n=l

n=l

IIpHMeHHH 2-A npH3HaK cpaBHeHHH, CpasHHM 3TOT pH,D; C paCXO)J;HIII.HMCH

00

1

rapMOHHqeCKHM PH,D;OM

En:

n=l

: 1) = lim

= -2 # o.

lim

(

1

1

n-+oo

2n -

In n

n

n-+oo 2 _

In n

1

n

an = 1 >

1

=an +l.

2n -Inn

2(n + 1) -In(n + 1)

r

r

1

r

1

= lim

1

= Q= O.

n

n

n~~ an = n~~ 2n -

In n

= n~~ 2n - In n

n-too 2 _

Inn

2

n

n

00

(-I)n+!

IITaK, .D:JIH ,n:aHHOrO 3HaKO'lepe.n:yIOlllerocHpH,n:a 2:

2

1

BbInOJIHeHbI

n=l

n -

nn

an+l

HHe ---a;;-:

an+! = n + 1 : ..!!.. = n + 1 . ~ = (1 + 1) .1.

an 3n+! 3n

n

3n+!

n 3

00 n

TIo npH3HaKY ,I1;arraM6epa OTCIO,n:a CJIe.u.yeT, 'ITOPM n~l 3n

CXO,n:HTCH, a 3Ha-

'1HT,HCXO,n:HbliI: PH,n: CXO,n:HTCH a6COJIIOTHO.

•

n

n

00

1

", .

L..J SIll 2"

n=l

n

PH,ll;OM

f: -\. 0603Ha'"lHBt

= -\ H Y'"IHTbIBM,'"ITO t

-+ 0 npH n -+ 00,

n=l n

n

HMeeM:

lim

(Sin - \ : -\)

= lim Sin t =

1 (l-ii

3aMe'"laTeJIbHbliinpe,ll;eJI).

n-+oo

n

n

t-+O

t

00

1

00

1

npH p = 2 > 1, TO

TaK KaK pH,ll;

I:

2" CXO,ll;HTCH KaK pH,ll; ,nHpHXJIe

I:

p

n=l n

1

n=l n

00

CXO,ll;HTCH H pH,ll;

I:

sin 2"'OTCIO,ll;a CJIe.n:yeT, '"ITOHCXO,ll;Hblii pH,ll; CXO,ll;HTCH

n=l

n

a6COJIIOTHO.

•

1.2.5.

IIccJIe,ll;oBaTb Ha CXO,ll;HMOCTb pH,ll;

~(_I)n+11. 4·7· ...

· (3n -

2).

a PacCMOTPHM pH,ll;

~

3·5·7· ...

· (2n + 1)

00

I: an H3 a6COJIIOTHbIX BeJIH'"IHH'"IJIeHOB,ll;aHHOrO pH,ll;a,

OTCIO,ll;a

3n + 1

3 + 1

1·

an+1

l'

l'

3 > 1

n

1m

-- =

1m

--- =

1m

-- = -

.

n-+oo

an

n-+oo

2n + 3

n-+oo 2 + 1

2

00

n

00

Ho 9TO 3Ha'"lHT,'"ITOpH,ll;

I: an pacXO,ll;HTCH, T. e. pH,ll;

I: (-1 )n+1 an He HBJIH-

n=l

n=l,

eTCH a6cOJIIOTHO CXO.lJ:jJrn;HMCH. 0

.

an+1

=

,ll;HaKO nOJIY'"IeHHbliipe3YJIbTaT ( hm -a-

n-+oo

n

3

)

an +1

>

= 2"

> 1

n03BOJIHeT C,ll;eJIaTb 60JIee CHJIbHOe YTBep)K,ll;eHHe. TaK KaK ~

> 1 ,lI,JIH Bcex HOMepoB n, Ha'"lHHMC HeKOToporo, TO an -+t

0 (n

-+

00), H

00

HCXO,ll;Hblii PM I: (-1)n+1 an pacXO,ll;HTCH.

•

n=l

n 2 + 1

1.2.6.

IIccJIe,ll;oBaTb Ha CXO,ll;HMOCTb pH,ll;

00

I: (_I)n

2

•

n=l

5n

- 2

a HeTPY,ll;HO nOKa3aTb, '"ITO,lI,JIH ,ll;aHHoro pH,ll;a He BbIllOJIHeH Heo6xo,ll;HMblii

npH3HaK CXO,ll;HMOCTH. B

caMOM ,ll;eJIe:

lim an =

.

n 2 + 1

= lim

1 + -\

1

~ O.

hm

-----.!!....2

= -5

n-+oo

n-+oo 5n2 - 2

n-+oo 5 __

n 2

CJIe,ll;OBaTeJIbHO, pH,ll; pacXO,ll;HTCH.

•

1.2.11. E(_1)n-1~.

00 (-l)n

1.2.12. L: (

)1·

n=l

n

n=l 3n -

2 .

1.2.13. E(_l)n+1 . _1_2 - .

n=l

n1n

n

00

1

1.2.14. L: (_1)n-1

n.

n=l

3·5·7· ... ·(2n+1)

n=l

1.2.16.

E(-1) n+1 3 . 7 ..... (4n -

1) .

n=l

5 . 8 ..... (3n + 2)

1.2.17.

~ (_1)n

3n2 -1 .

1.2.1

8

.

L..J

2

n=l

5 + 2n

1) iJM 'H.e06xoiJuMoao npU3'H.a'ICa -

lim an;

n-too

2) iJM 1-ao U 2-ao npU3'H.a'IC06 Cpa6'H.e'H.USI -

06w,uiJ. "t.II.e'H. pSliJa, C 'lCOmOp'btM

Cpa6'H.U6aemCSl

iJa'H.'H.'btiJ. PSliJ;

'I

iJ

.:

1·

an +1

3;

.II.SI npU3'H.a'ICa ,l(a.ll.aMuepa -

1m

-a-;

n-too

n

4) iJM npU3'H.a'ICa KoutU -

lim

Viani.

n-too

E(_1)n 3n + 1.

00

(_1)n-1

1.2.20.

1.2.19.

n~l

en+1

n=l

3n-1

1.2.21.

E

1.2.22. E(_1)n-1_1_.

(_1)n-1 (2nn\!.

n=l

4 n.

n=l

3n-1

1.2.23.

E(-1)n1nn (

2n 2 ).

1.2.24.

00

•

L: (_1)n

2 .

n=l

n+

Sln n

n=l

n

1.2.25. E(_1)n+1_1_.

1.2.26.

n=2

ny1nn

00

2

1.2.27.

L:(_1)n-1 nn .

n=l

2

00

(3 + i)n

1.2.28. I1ccJIe,n:oBaTb Ha CXO,n:HMOCTb PM n~l

n. 2n

n·2n

(3 + i)nH I

n2n

= 13 + il

n

= I (3 + i)n . (n + 1)2nH

- 2 - . n + 1 '

oTKy,n:a

. _n_) = 13 + il

n

3

2

lim

+ il

= v'3

+ 12

= VIO > l.

Ia+l I= lim (1

n-+oo

an

n-+oo

2

n + 1

2

2

2

CJIe,n:OBaTeJIbHO,

Ia~:l I > 1 ,n:JIH Bcex HOMepoB n,

HaqHHaH C HeKOTOpOI'O,

oTKy,n:a

lim an i:- 0, H 3HaqHT, HCXO,n:Hblii pH,n: pacXO,n:HTCH.

•

n-+oo

1.2.29.

I1ccJIe,n:oBaTb Ha CXO,n:HMOCTb PH,n: f: ((

n ~3i

) n

n=l

2

+ z n + 1

PMa npH3HaK KOillH. CHaqana npe06pa3yeM BbIproKeHHe

Viani:

\IjaJ = n I(

n+3i

)nl

-

I

n+3i

I

In+3il

(2 + i)n + 1

(2 + i)n + 1 -

1(2n + 1) + inl -

1+ Jl

V(2n + 1)2 + n 2

5n2 + 4n + 1

n 2

OrcIO,n:a

lim

~= lim

n-+oo

n-+oo

00

TaKHM 06pa30M, PM 2: lanl

CXO,n:HTCH, T. e. HCXO,n:Hblii pH,n: CXO,n:HTCH a6co-

n=l

•

JIIOTHO.

·n

00

1.2.30. I1ccJIe,n:oBaTb Ha CXO,n:HMOCTb PH,n: 2:

zr;;;.

Q 1. TIOCKOJIbKY I~I= ~ = In'

n=l yn

TO PH,n:, COCTaBJIeHHblii H3 a6COJIIOT-

00

1

HbIX BeJIHqHH qJIeHOB ,n:aHHOI'OpH,n:a, HMeeT BH,n:

2:

r;;;.

TIOJIyqeHHbIii PH,n:

pacXO,D;HTCH KaK pH,D; ,il;HpHXJle f:

\ npH p =

1

-2 < 1. 3HaqHT, HCXO,D;HbIii

n=l n

pH,D; He HBJIHeTCH a6COJIIOTHO CXO,lJ;jJIII.HMCH,

2,

3anHilleM qJIeHbI ,D;aHHOrO pH,D;a B aJIre6paHQeCKOii q,opMe, T, e, B BH,D;e

bn + icn :

00

'n

'

'2

'3

'4

'5

'6

'7

'8

L

_1_ =

1.

+ _1_

+ _1_

+ _1_ + _1_ +

_1_ + _1_ + _1_ + .. , =

n=lvn

1

y'2

vra

J4

y'5

v'6

y'7

VB

iii

1

iii

1

=1- y'2-

vra+ J4+ y'5- v'6- y'7+VB+"'=

=(O+i)+(- ~+Oi)+{O- Ja)+(~+Oi)+(O+ Jg)+

+ (- ~+Oi) + (0- ~) + (Js+Oi) + .. ,

00

00

COCTaBHM ,D;Ba pH,D;a E bn H

E iCn

COOTBeTCTBeHHO H3 ,D;eiiCTBHTeJIbHbIX H

n=l

n=l

MHHMbIX qacTeii qJIeHOB nOCJIe,D;HerO pH,D;a:

00

"

1

1

1

1

L..J bn = 0 -

In + 0 + .r;; + 0 -

IE + 0 +

10 + .. , ,

n=l

v2

v 4

v6

v8

00

L iCn = i + 0 - ~ + 0 + ~ + 0 - ~ + 0 + .. ,

n=l

v3

v5

v7

00

-~+ ~_~+~+ .. ,+(_I)n ~+"'=L(-I)nb~,

v2

v 4

v6

v8

v 2n

n=l

,

iii

i

00

(l)n-1

' "(

l)n-1,

1 -

. r.;

+

.~ -

. r-; + ' , ,+ -

~

+ ' ,, = 1 'L..J -

Cn '

v 3

v 5

v 7

v 2n - 1

n=l

00

00

,Il;JIH 3HaKOqepe,D;yIOIII.HXCH PMOB E (-I)nb~ H

E (_I)n-1c~ BblnOJIHHIOTCH

n=l

n=l

06a yCJIOBHH npH3HaKa JIeii6HH~a, TaK KaK npH Bcex n = 1,2,3, ' "

cnpase.u,-

JIHBbI COOTHOIIIeHHH

b'

+1

=

lib'

I'

b'

I'

1

=

0

,

n

v2(n + 1)

< .rn= = n H

1m

n =

1m . rn=

v 2n

n-+oo

n-+oo V

2n

, _

1

1

1

'1"

Cn =

0

,

cn+1

-

v2(n+l)-1

= Jl'C""'i"'"

< ~ = cn H

1m

v2n+1

v2n-1

n-+oo

00

00

3HaQHT, pH,D;bI E (-I)nb~

H E (-I)n- 1c'n

CXO,lJ;jJTCH, T,e,

CXO,lJ;jJTCH

PMbI

00

00

n=l

n=l

00

'n

00

00

n~lbn H n~lCn' OTCIO,D;a CJIe,D;yeT, QTO pH,D;

n~l

jn

= n~lbn + i n~lCn CXO-

1) a.II..R He06xoau.M.oeo npU3Ha7&a -

lim an;

n-+oo

2) a.II..R 1-eo U 2-eo npU3Ha7&06 cpa6HeHUSI -

06w,uit

",.n.eH pSlaa, c 7&omop'bl..M.

cpa6HU6aemC.R aaHH'bI.it pSla;

3) a.II..R npU3Ha7&a ,[(a.n.a.M.6epa -

lim

a~+1;

n-+oo

n

4) a.II..R npU3Ha7&a Kowu -

lim

Viani.

n-+oo

00

n

e+if·

1.2.31.

~

1.2.32.

E

n=l 2n +i..jii

n=l

3

00

1

00

·2n

n~l

~_z.

1.2.33.

n(2 +i)n

1.2.34.

n=l ..jii

1.2.35.

00

(2n+i f

1.2.36.

E

(3-if·

n~l

3ni - 2 .

n=l

2

1.2.37.

E

cos n +isin n

n=l

n2

•

,[(07&a3am'b,

",mo pSla CXOaumC.R YC.n.06HO:

1.2.38.

00

(_I)n

1.2.39.

E(-I)n

1

~

.

n=l

..jii + 2

n=l

n Vlnn + 2

1.2.40.

E

(_I)nlnn.

1.2.41.

E(_I)n+1 n2

+ 3 .

n=l

..jii

n=l

n

+ 4

1.2.42.

E(_I)n

n + 1

1)

n=l

1·3·5· .... (2n -

1.2.43.

E(-I)n+ltg

Ir.:;-.

1.2.44.

n=l

nyn