Сборник задач по высшей математике 2 том

Hcc.n.eaoeam'b pSia Ha CXOa'U.M.OCm'b:

1.2.50.

E(_l)nH n + 2 .

1.2.51.

00

(-1)n- 1n

n~l (2n + 1) . 3n ·

n=l

2n + 5

1.2.52.

E(-l)n :+2 .

1.2.53.

n

E(_1)n+13 : .

n=l

3n

vn

+ 1

n=l

n

1.2.54.

E(_l)nH 3n

1.2.55.

~ (_

)n cos(n + 2)

.

L..J

1

3n

n=l

n(n + 1)

n=l

(_1)nn2

1.2.56.

00

(_l)n-l

1.2.57.

00

~

n(2 + In n)

3·

~ -'---==---

n=l

n=l nvn+3n

1.2.58.

E(_l)n-l ln (nn+ 1).

1.2.59.

E

(i(n+2i))n

n=l

n(2 + i)n

n=l

3n

1.2.60.

00

1.2.61.

00 i+(-l)n.n

n~l

3n

~

n

2

•

n=l

1.2.62.

00

n(l + i)n

1.2.63.

00

1

n~l

3n

n~l

(n +i)vn

00

·n

00

•

1.2.64.

~ ~.

1.2.65.

~ cosm

n=l

3n .

n=l

00

n

.

1.2.66.

~

n=l

vn +in

KOHTponbHble Bonpocbl

...

60nee CnO)KHble 3aACIH...

"

1.2.67.

BepHO JIH, '"ITO

a) eCJIH PM a6COJIIOTHO CXO,D;HTCH, TO OH CXO,n;HTCH H YCJIOBHO;

6) ecJIH pH,D; CXO,D;HTCH YCJIOBHO, TO OH He CXO,D;HTCH a6cOJIIOTHO?

1.2.68.

00

n 2

+ n

n

IIccJIe,D;oBaTb Ha CXO,D;HMOCTb PM n~l

( -1) - 2 - . 2n ·

1.2.69.

IIccJIe,D;oBaTb Ha CXO,D;HMOCTb pH,D; E(-1) n

sh n

n=l

Vch2 n + 1

00

1.2.70.

BepHo JIH, '"ITOeCJIH 3HaKOnepeMeHHblii pH,D;

~ (-l)nan CXO,D;HTCH,

TO an -+ 0 (n -+ 00) MOHOTOHHO?

n=l

1.2.71.

BepHO JIH ,n;JIH 3HaKOnepeMeHHoro pH,D;a, '"ITO

00

a) eCJIH nOCJIe,D;OBaTeJIbHOCTb an MOHOTOHHa, TO pH,D; ~ an . (-1)n

CXO,D;HTCH;

00

n=l

6) eCJIH an -+ 0 (n -+ 00), TO pH,D; ~ (-l)na n CXO,D;HTCH;

n=l

00

B) eCJIH an -+ 0 (n -+ 00) MOHOTOHHO, TO pH,D;

~ (-l)nan CXO,n;HTCH

YCJIOBHO;

n=l

00

r)

eCJIH an

-+ 0 (n -+ 00)

MOHOTOHHO, TO pH,D;

~ (-1) nan CXo-

,D;HTCH.

n=l

30

1.2.72.

,IJ;oKa3aTb

)J,JUI

3HaKoIIepeMeHHblx

PH,Il;OB

CJIe.n.yIOm;He yTBep:>K,Il;e-

HHH:

a) PM CXO,Il;HTCH a6COJIIOTHO TOr,Il;a H TOJIbKO TOr,Il;a, KOr,Il;a CXO,Il;HT-

CH ,Il;Ba pH,Il;a - pH,Il; H3 IIOJIO:>KHTeJIbHbIX '1JIeHOBH PM H3 OTpHIIa-

TeJIbHbIX'IJIeHOB;

6) eCJIH pH,Il; CXO,Il;HTCH YCJIOBHO, TO pacXO,Il;HTCH ,Il;Ba pH,Il;a -

PM

H3 IIOJIO:>KHTeJIbHbIX '1JIeHOBH pH,Il; H3 OTpHIIaTeJIbHbIX '1JIeHOB;

u) eCJIH O,Il;HH H3 ,Il;BYX PH,Il;OB (C IIOJIO:>KHTeJIbHbIMH '1JIeHaMHH OT-

pHIIaTeJIbHbIMH '1JIeHaMH)CXO,Il;HTCH, a ,Il;Pyroti -

pacXO,Il;HTCH, TO

HCXO,Il;Hblti pH,Il; pacXO,Il;HTCH.

00

1.2.73.

ECJIH pH,Il;

E an CXO,Il;HTCH YCJIOBHO, 'ITOMO:>KHO CKa3aTb 0

CXO,Il;H-

n=l

MOCTH pH,Il;a H3 ero IIOJIO:>KHTeJIbHbIX '1JIeHOB?

1.2.74.

lICCJIe,Il;OBaTb pH,Il; Ha CXO,Il;HMOCTb:

{-k,

1

1

1

1

n -

'1eTHoe;

a) 1 -

2 + 32 -

4" + 52

-

... an =

_l

n -

He'leTHoe.

n 2 '

1 1

1

1

1

In = 2k -1;

{

- 2k - 1 '

6) 1 -

3 + 2 - 32 + 22

-

35 + ... an =

I_

n = 2k.

32k - 1 '

u) 1 -

11111

1

1

3 +

3 -

32 +

5"

-

33 + ... , a2k-l

= 2k _

l' a2k = - 3k '

r)

1

1111

1

1

3'

3 - 1 +

'7- 5" + 11 -

9 + ' . " a2k-12 =

4k - l' a2k ;:: -

4k _

~)

1____1_+

1____1_+ ..., a2k-l =

1

,

v'2 - 1

v'2 + 1

v'3- 1 v'3+ 1

Jk + 1 -

1

a2k =

1

Vk+l+t'

00

1.2.75.

,IJ;oKa3aTb,

'ITO ecJIH

pH,Il;

E an

CXO,Il;HTCH a6COJIIOTHO,

TO

pH,Il;

00

n

+ 1

n=l

E

-n-an CXO,Il;HTCH a6cOJIIOTHO.

n=l

00

00

1.2.76.

,IJ;oKa3aTb, 'ITOeCJIH

pH,Il;bI

E a;

H E b; CXO,Il;HTCH a6COJIIOTHO,

00

n=l

n=l

TO pH,Il; E anbn CXO,Il;HTCH a6cOJIIOTHO.

n=l

1.2.77. ,IJ;oKa3aTb, 'ITOecJIH pH,Il; CXO,Il;HTCH a6COJIIOTHO, TO H PM, IIOJIY- '1eHHblti H3 HCXO,Il;HOrO C IIOMOlllhIO IIPOH3BOJIbHOti IIepecTaHoBKH ero '1JIeHOB,TaK:>Ke CXO,Il;HTCH a6cOJIIOTHO, "pH'IeMK TOti :>Ke CYMMe, 'ITOH HCXO,Il;Hblti PM.

1.2.78.Teope.M.a P'U.MaHa. ,IJ;oKa3aTb, 'ITOecJIH pH,Il; CXO,Il;HTCfl YCJIOBHO,

TO cym;ecTByeT TaKaH IIepecTaHoBKa ero '1JIeHOB,'ITOIIOJIY'leHHblti

pH,Il; CXO,Il;HTCH K JII060MY HaIIepe,Il; 3a,IJ;aHHoMY '1HCJIYHJIH pacxo-

,Il;HTCH 3a,IJ;aHHbIM 06pa30M (K +00, K -00 HJIH K 00).

31

§ 3. CTEnEHHblE PHAbl

(3.1)

r,D;e ao, a I, a2, ... , an, . .. - nOCTOfiHHble '1HCJ1a(,D;eil:cTBHTeJIbHble HJIH KOMnJIeKCHble), a x - nepeMeHHafi BeJIH'IHHa(TaKJoKe ,D;eil:cTBHTeJIbHM HJIH KOMnJIeKCHM), Ha3b1BaeTCfi cmeneHH'bI.M PRiJOM. QHCJ1a ao, a I, a2, .•. , an, . .. Ha3b1BaIOTCfi IC03tP-

tPU'4UeHmaMU cmeneHHOZO pRiJa. COKpaw;eHHO CTeneHHoil: pfi,D; 0603Ha'laIOT TaK:

~BY,D;eM Ha3b1BaTb cTeneHHoil: PM iJei1,cm6UmeJlb'lt'bl.M (COOTBeTCTBeHHO, ICOMn-

JleICCH'bI.M) cmeneHH'bI.M PRiJOM, eCJ1H ero K094><lmIJ;HeHTbI - ,D;eil:cTBHTeJIbHble (COOTBeTCTBeHHO, KOMnJIeKCHble) '1HCJ1a, a nepeMeHHM x npHHHMaeT ,D;eil:cTBHTeJIbHble

(COOTBeTCTBeHHO, KOMITJIeKCHble) 3Ha'leHHfi.	~
QacTO paccMaTpHBaIOT CTeneHHble pfi,D;bI 60JIee 06w;ero BH,D;a
00
L an(x - at = ao + aleX - a} + a2(X - a}2 + ... + an(x - at + ... ,	(3.2)
n=O

'1aCTHbIMCJ1Y'laeMKOTOPbIX npH a = 0 fiBJIfiIOTCfi 06b1'1HbleCTeneHHble pfi,D;bI (3.1).

C ,D;pyroil: CTOPOHbI, KaJoK,D;bIiI: cTeneHHoil: pfi,D; BH,D;a (3.2) C nOMOIIJ;bIO 3aMeHbI nepe-

00

MeHHoil: y = x - a CBO,D;HTCfi K pfi,D;y E anxn BH,D;a (3.1). n=O

~IIpH,D;aBM nepeMeHHoil: x B CTeneHHOM PMe KOHKpeTHoe '1HCJ10BOe3Ha'leHHe

x = Xo, nOJIy'lHM'1HCJ10BOiI:PM, KOTOPblil: CXO,D;HTCfi HJIH pacxO,D;HTCfi. MHOJoKeCTBO Bcex Tex 3Ha'leHHiI:nepeMeHHoil:, npH KOTOPblX ,D;aHHbliI: cTeneHHoil: pfi,D; CXO,D;HTCfi,

Ha3b1BaeTCfi 06J1aCmb'lO cxoiJUMocmu 9Toro pfi,D;a. ~

IIpH X = 0 (COOTBeTCTBeHHO, npH x = a) BCfiKHiI: cTeneHHoil: PM BH,D;a (3.1)

(COOTBeTCTBeHHO, BH,D;a (3.2}) CXO,D;HTCfi, n09TOMY 06JIacTb CXO,D;HMOCTH CTeneHHoro pfi,D;a CO,D;epJoKHT no Kpail:Heil: Mepe O,D;HY TO'lKy.

00

TeopeMa 1.5 (A6enR). ECIlIll CTeneHHoA p1lA E anXn CXOAIIITC1I B TO'lKeXo. TO n=O

OH a6COlltOTHO CXOAIIITC1I B Ka>KAOA TO'lKeX. AIl1l KOTOpoA Ixl < Ixol.

00

CnllACTBlile 1.1. ECIlIll CTeneHHoA P1lA E anxn paCXOAIIITC1I nplll HeKOTopOM 3Ha- n=O

'1eHIIIIIIx = Xl. TO OH paCXOAIIITC1I III nplll BCex 3Ha'leHIII1IXX. AIl1l KOTOPblX Ixl > IXII.

~ HHmep6aJ10M CXoiJuMocmu ,D;eil:cTBH1IeJIbHOrO CTeneHHOrO pMa BH,D;a (3.1) (coOTBeTCTBeHHO, BH,D;a (3.2}) Ha3b1BaeTCfi TLwil: HHTepBaJI (-R, R) (COOTBeTCTBeHHO,

(ao - R,ao + R)), 'ITOB KaJoK,D;Oil: ero TO'lKepfi,D; CXO,D;HTCfi a6COJIIOTHO, a B KaJoK,D;Oil:

32

CTBeHHO, BH,D;a (3.2»

TO'lKe,JIe:lKam:eii BHe OTpe3Ka [-R, R] (COOTBeTCTBeHHO, [xo-R, xo+R]), pH,D; pacxo- ,D;HTCH. Ha rpaHHU;ax HHTepBaJIa CXO,D;HMOCTH, T. e. B TO'lKaxx = ±R (COOTBeTCTBeHHO, B TO'lKaxx = Xo ± R), PM MO:lKeT KaK CXO,D;HTbCH, TaK H pacxO,D;HTbCH. QHCJ10

R Ha3blBaeTCH paiJUYCOM cxoiJuMocmu ,D;eiiCTBHTeJIbHOrO CTerreHHoro pMa. ~

B '1acTHOCTH,R MO:lKeT PaBHHTbCH HyJIIO - B 3TOM CJIy'lae06JIacTb CXO,D;HMOCTH plI,D;a COCTOHT H3 O,D;HOii TO'lKH0 (COOTBeTCTBeHHO, XO), HJ1H +00 - B 3TOM CJIy'lae 06JIacTbIO CXO,D;HMOCTH lIBJIlIeTClI BClI '1HCJ10BalI rrplIMaH (TaKOii plI,D; Ha3blBaeTClI em:e 6C'lOiJy cXOiJ.RU4UMCJI).

~KpyeoM cxoiJUMocmu KOMIIJ1eKCHOro CTerreHHoro plI,D;a BH,D;a (3.1) (cooTBeT-

Ha3blBaeTClI TaKoii OTKPblTblii Kpyr Ixl < R (cooTBeTcTBeHHo,

Ix-al < R), 'ITOB Ka:lK,D;Oii ero TO'lKepH,D; CXO,D;HTClI a6COJIIOTHO, a B Ka:lK,D;oii TQ'IKe, JIe:lKam:eii BHe 3aMKHYToro Kpyra Ixl ~ R (COOTBeTCTBeHHO, BHe 3aMKHYToro Kpyra

Ix - al ~ R), plI,D; pacXO,D;HTClI. ~

B rpaHH'IHblXTO'lKaxKpyra CXO,D;HMOCTH - T.e. Ha OKPY:lKHOCTH Ixl = R (coOTBeTCTBeHHO, Ix - al = R) - plI,D; MO:lKeT KaK CXO,D;HTbClI, TaK H pacXO,D;HTbCH. QHCJ10 R Ha3blBaeTCH paiJuycoM cxoiJUMocmu KOMIIJ1eKCHOro CTerreHHoro plI,D;a. B

'1aCTHOCTH, R MO:lKeT 6blTb PaBHO 0 - B 3TOM CJIYQae BClI 06JIacTb CXO,D;HMOCTH

plI,D;a COCTOHT H3 O,D;Hoii TO'lKH0 (COOTBeTCTBeHHO, a), HJIH +00 - B 3TOM CJIy'lae 06JIacTbIO CXO,D;HMOCTH HBJIlIeTClI BCH KOMIIJ1eKCHaH IIJ10CKOCTb C.

HHTepBaJI H Kpyr CXO,D;HMOCTH pH,D;a, KaK rrpaBHJ10, orrpe,D;eJIlIIOT C rrOMom:bIO rrpH3HaKa )J;aJIaM6epa HJ1H rrpH3HaKa KomH, rrpHMeHeHHblX K 3HaKOrrOJIO:lKHTeJIb-

HOMY plI,D;y

00	00
L lanxnl (COOTBeTCTBeHHO, L lan(x - at!),
n=O	n=O

COCTaBJIeHHOMY H3 a6COJIIOTHblX BeJIH'IHH'lJIeHOBHCXO,D;HOrO CTerreHHoro pH,D;a. )];JIH Bbl'lHCJ1eHHlIpa.n;Hyca CXO,D;HMOCTH R CTerreHHoro PMa rrpHMeHlIIOTCH TaK-

:lKe <P0PMYJIbl:

R- lim

I~I

H R= _---=1~=

-

n--+oo

an +l

lim

Viani

n--+oo

B TeX CJIy'lalIX,KOr,D;a YKa3aHHble rrpe,D;eJIbI cym:eCTBYIOT.

00

n!(x _ 3)n-1

1.3.1.

HaitTH 06JIacTb CXO,lI,HMOCTH pH,lJ,a

E

n+1

n=l

2

a TIpHMeHHM npH3HaK ,I1;aJIaM6epa. TIOCKOJIbKY

1

an+1 1 = I(n + 1)!(x - 3)(n+1)-1 :

n!(x -

3)n-1 I=

an

2(n+1)+1

2n+1

(n + 1)!.

(x -

3)n

. 2n +11 = I(n + 1). (x _ 3) . 11 = Ix -

31

. (n + 1)

I

n!

(x -

3)n-1

2n+2

2

2

'

TO

x'" 3,

1.

1a +

1

1_ 1·

Ix - 31 (

1) _ {+oo npH X -

3 '"0,

n

-

°npH x - 3 = 0,

lID --- 1ID---·n+

x = 3.

n-too

an

n-too

2

2 C60pHHK _~ no aweweA wareM8THKC. 2 KYJlC

33

TaKHM 06pa3oM, p»,II. CXO,Il.HTC» (a6COJIIOTHO) TOJIbKO npH x =						3, B OCTaJIbHb~
TO'lKax:'1HCJIOBOilnp»Moil		P».II. pacXO,n;HTC».		00 3n- 1(x+l)n			•
1.3.2.
	HailTH 06JIacTb CXO,Il.HMOCTH p»,Il.a			E	n
a BOCnOJIb3yeMc» npH3HaKOM Korrrn:				n=1	n
r	n~lfIl- r n	1 3n- 1(x+l)nl_ r			Ix+ll 3 n - 1		_
n~	Vlanl- n~	nn	-	n~-n--'		n	-
	1-1.
=lx+lllim 3 n n		=lx+ll·0=O<1			npHBcexxE(-oo,+oo).
	n--+oo

CJIe,n;OBaTeJIbHO, P».II.	CXO,Il.HTC» a6COJIIOTHO B KIDK,n;oil TO'lKe'1HCJIOBOilnp»-
Moil (-00, +00).	•

00

1.3.3.HailTH 06JIacTb CXO,Il.HMOCTH p».II.a E xn.

a IIpHMeHHM npH3HaK )1;aJIaM6epa:								n=1
		lim	-a-		=	lim	Ixn+11 = lim Ixl = Ixl·
			an +1	1			- n -
		n--+oo l n				n--+oo		n--+oo
							X
(9TOT :>Ke		pe3YJIbTaT		MO:>KHO			nOJIY'lHTb, npHMeH»» npH3HaK KOillH:
lim	Viani =	lim	Vlxnl		=	lim Ixl =		Ixl.) OTcIO,n;a CJIe.n;yeT, 'ITOnpH
n--+oo		n--+oo				n--+oo
Ixl <	1 (T.e. npH x E		(-1,1»		p»,n; CXO,Il.HTC» a6cOJIIOTHO, npH Ixl > 1 pacxo-
,Il.HTC». TaKHM 06pa30M, HHTepBaJI (-1,1) -								HHTepBaJI CXO,n;HMOCTH ,n;aHHOrO

p».II.a. lIccJIe.n;yeM P».II. Ha CXO,Il.HMOCTb B rpaHH'IHblXTO'lKax:9TOro HHTepBaJIa, T. e. B TO'lKax:x = -1 H x = 1.

IIp~ x = -1 nOJIy'lHM3HaKO'lepe.n;yIOIII,Hilc»p»,II.

00

L(-I)n = -1 + 1-1 + 1- ... + (_I)n + ...

n=1

9TOT P».II. pacXO,Il.HTC», T.K. He BblnOJIHeH He06XO,Il.HMbIil npH3HaK CXO,Il.HMOCTH

(an -f+ 0 npH n --t 00).						~
IIpH X = 1 nOJIy'lHMp»,n;
	00
	L 1n =I+I+I+ ... +I+ ...
	n=1
9TOT P».II. pacXO,Il.HTC» no Toil ,:>Ke npH'IHHe,TaK KaK
	lim	an = lim 1 = 1 =I- O.
	n--+oo	n--+oo				•
lITaK, 06JIacTb CXO,Il.HMOCTH ,n;aHHoro p»,n;a -			HHTepBaJI (-1,1).
			00	(x _ 2)n+1
1.3.4.	HailTH 06JIacTb CXO,n;HMOCTH p»,n;a		E	n (	)
			n=1	3 n	+ 2

34

o 1. IIpHMeHHM npH3HaK ,I:(a.rraM6epa. YqHTbIBaJI, 'ITO

an+l	I(X - 2)(nH)H			(X - 2)n+l I
i--u;;-i =		3nH (n + 1 + 2) :		3n(n + 2)	=
			= 1(x - 2)n+2 .		~ . n + 21 = Ix -			21 . n + 2
			(x - 2)nH		3nH	n + 3	3	n + 3'
lim ian +1		i	Ix - 21	n+ 2	Ix - 21	. lim	n + 2	Ix-21
			J~~--3-· n+3		3	n-+oo n + 3		- 3 -
n-+oo	an
OTCIO,ll;a
Ix -	21		x -	2	< x -	2 < 3 ¢:} -1 < x < 5.
- 3 -		< 1 ¢:} -1 < -3- < 1 ¢:} -3

HTaK, npH X E (-1,5) PH)]; CXO,ll;HTCg a6cOJIIOTHO, a npH x f/. [-1,5)- pacxo- ,l1,HTCg. 3HaqHT, (-1,5) - HHTepBa.rr CXO,ll;HMOCTH ,l1,aHHoro pg,ll;a. HCCJ1e.rr.yeM CXO,ll;HMOCTh pg,l1,a Ha KOHU;ax 9TOro HHTepBa.rra, T. e. B TOqKax x = -1 H x = 5.

2. IIpH X = 5 nOJIyqHM PH)];
00	(5 _ 2)nH	00 3n +1	00 3
~	3n(n+2)	= ~ 3n(n+2) =	~ n+2·

IIpHMeHgg 2-it npH3HaK cpaBHeHHg, cpaBHHBaeM 9TOT PH)]; C rapMOHHqeCKHM

PH)];OM fit: n=l

lim	(_3_: 1) = lim			~ =		lim _3_ =3#0.
n-+oo	n	+ 2 n	n-+oo n + 2			n-+oo 1 +	1
		1					n
IIocKOJIhKY PH)];	00
	L n pacXO,ll;HTCg, a nOJIyqeHHhIit npe,ll;eJI He paBeH HyJIIO,
	n=O
TO PH)]; f --L2 pacXO,ll;HTCg.
n=on+
3. IIpH X = -1 nOJIyqHM PH)];
00 (-1-2)nH		00 (_3)nH		00	(_1)n+l.3nH		00	3
~ 3n(n+2)	= ~ 3n(n+2) =			~	3 (n+2)		= ~(_1)n+ln+2.
						n
OTOT PH)]; He gBJIgeTCg a6cOJIIOTHO CXO~Iu;HMcg, TaK KaK pg,l1, f								+3 2'
							n=on

COCTaBJIeHHhIit H3 a6cOJIIOTHhIX BeJIHqHH qJIeHOB ,l1,aHHOrO pg,ll;a, pacXO,ll;HTCg (CM. nyHKT 2).

BhIgCHHM, CXO,ll;HTCg JIH ,ll;aHHbIit 3HaKOqepe.rr.yIOID;HitCg PH)];, HCnOJIh3yg npH3HaK JIeit6HHu;a.

a) OqeBH,ll;HO, HepaBeHCTBO

3 >	3	-a
an = n + 2	(n + 1)	+ 2 - nH

BhInOJIHeHO ~ Bcex n = 1,2, ...

35

6) KpoMe TOro,	an = lim	~2 = O.
lim
n-+oo	n-+oo n +
IhaK, ,!I,JUI 3HaKO'iepe~IOlll;erocH pH)J;a		f: (-1) n+1 ~2 BhmOJIHeHhI 06a
		n=O	n+

YCJIOBHH, cO)J;ep)Kalll;HeCH B npH3HaKe JIeit6HHu;a, 3Ha'iHT,)J;aHHhIit pH)J; CXO- )J;HTCH. TaK KaK OH He HBJIHeTCH a6COJIIOTHO CXOMlll;HMCH, TO pH)J; CXO,!I;HTCH YCJIOBHO. OKOH'iaTeJIhHOnOJIY'iHM,06JIacTh CXO)J;HMOCTH HcxO)J;Horo pH)J;a-

npOM~YTOK [-1,5).						00 (x + 5)n			•
1.3.5.
	HaitTH 06JIacTh CXO)J;HMOCTH pH)J;a					E		3 .
						n=2 3n+1n In		n
a 1. JIpHMeHHM npH3HaK ,Il,aJIaM6epa. TaK KaK
an+l I	I	(x + 5)n+1			(x + 5)n		I
I""'U;- = 3(n+l)+1(n+l)In3(n+l): 3n+1nIn3n							=
(x + 5)n+l		3n+1	n	In3 n	I	Ix + 51		n	In3 n
=I (x + 5)n		. 3n+2 . n + 1 . In3(n + 1)				= - 3 - . n + 1 . In3(n + 1)'

TO

lim Ian +1 1 = lim

n-+oo an

n-+oo

(IX + 51) . _n_ .		In3	) =
3	n + 1	In3 (n + 1)

= Ix + 51 . lim _n_. lim In3 n	= Ix + 51 .1 . 1 =	Ix + 51.
3 n-+oo n + 1 n-+oo In3 (n + 1)	3	3

(JIPH BhI'iHCJIeHHHnOCJIe)J;Hero npe)J;eJIa BOCnOJIh30BaJIHCh paBeHCTBaMH

lim In3 n	= lim	(	In n ) 3 =	[ lim In n	] 3
n-+oo In3(n + 1)	n-+oo		In(n + 1)	n-+oo In(n + 1)
H, )J;aJIee, npaBHJIOM JIonHTaJIH.) Hait)J;eM HHTepBaJI CXO)J;HMOCTH
Ix+51	x+5	< 1 ¢:} -3 < x + 5 < 3 ¢:} -8 < x			< -2.
- 3 - < 1 ¢:} -1 < - 3 -

IhaK, npH x E (-8, -2) pH)J; CXO)J;HTCH a6COJIIOTHO. llCCJIe~eM CXO,!I;HMOCTh pH)J;a B TO'iKaXx = -8 H x = -2.

2. JIPH x = -8 nOJIY'iHMpH)J;
00	(-8 + 5)n	00	(-3)n		00	3n	00	(_1)n
"	-	"	3n+1n In	-	"(-1)n	-	" --'----'::-
L...J	3n+1n In3 n -	L...J		3 n -	L...J	3n+1 n In3 n -	L...J	3n In3 n'
n=2		n=2			n=2		n=2

llCCJIe~eM 9TOT pH)J; Ha CXO)J;HMOCTh. PacCMOTPHM PH,!I;, COCTaBJIeHHhIit H3

a6cOJIIOTHhlX BMH'iHH'iJIeHOB)J;aHHOrO pH)J;a:

00

L 3nI~3n'

n=2

36

ilpHMeHHM HHTerpaJIbHhlit npH3HaK. TaK KaK an = 3nIn1 3 n , TO

O'leBH,lI,HO,'ITOf(x) MOHOTOHHO y6bIBaeT Ha npOMe:>KYTKe [2, +00), T. e.

VXI>X2>2=}f(XI)=	13	<	1	=f(X2)'
	3XI In	Xl	3X2 In3 X2

TaK KaK <PYHKlIHH f(x) nOJIO:>KHTeJIbHa, HenpepbIBHa H MOHOTOHHO y6bIBaeT Ha npOMe:>KYTKe [2, +00), TO ~H HCCJIe.II:OBaHHH .II:aHHOrO pH.II:a Ha CXO,!I,HMOCTb MO:>KHO npHMeHHTb HHTerpaJIbHbIit npH3HaK.

CHaqaJIa HaMeM HeOnpe.II:eJIeHHbIit HHTerpaJI

OTClO.II:a

+00		+00				M
f	f(x)dx ~	dx	=	lim		fdx =
		f 3x In3 X		M-Hoo		3x In3 X
2	= 11m	2				2		+ --1)
		1 IM) =				(1			=_1_.
	·	( ---			l'1m -		6In2 M
	M-Hoo	6In2 X	2		M--Hoo			6In2 2 6In2 2
					+00
TaK KaK Heco6cTBeHHbIit HHTerpaJI f						dX3	CXO,!I,HTCH, TO CXO.II:HTCH H pH.II:
					2	3xIn	X
00	1		00	(-I)n
~	3'a 3HaqHT, pH.II:		~		3	CXO,!I,HTCH a6COJIlOTHO.
n=23nIn n			n=23nIn			n
3. IIpH X = -2 nOJIyqHM pH.II:

(hOT pH.II: CXO,!I,HTCH a6COJIlOTHO (CM. nyHKT 2).
TaKHM 06Pa30M, 06JIacTb CXO.II:HMOCTH HCX0.II:HOrO pH.II:a -			npOMe:>KYTOK
[-8,-2].			•
1.3.6.	HaitTH Kpyr CXO.II:HMOCTH KOMnJIeKcHoro CTeneHHoro pH.II:a
	f: (2i)n+l(z ~ !i)n
	n=l	(v7-3z)

37

a llpHMeHHM npH3HaK KOillH:
I.	n~if'1-	I'	n 1(2i)n+1(z.+ 3i)n 1_ I'					I	+ 3'1 1(2i) n~11 -
1m	Viani -	1m		(.;7 - 3i)n		-	1m	z	z·	-
n-+oo		n-+oo					n-+oo		1.;7 -	3il
			1
	lim 12ill+n								Iz + 3il· 2
= Iz + 3il . .c:..n-+-,-oo.:...::::-__				= Iz + 3il .		12il
		1.;7 -	3il			1.;7 -	3il	J(.;7)2 + (-3)2
							=	Iz + 3il ·2 Iz + 3il
									4	2
Hait.n;eM Kpyr CXO.n;HMOCTH pH.n;a:
			Iz + 3il			.	< 2.
			2		< I ¢} Iz + 3z1
IhaK, B Kpyre Iz + 3il < 2 cTeneHHoit pH.n; CXO.n;HTCH a6cOJIIOTHO.										•

HaiJ.mu 06.1/,acm'b CXOOUMocmu pROa. Y1Ca3am'b npUMe1tJleM'bte npU31ta1CU. )1,0- nO.l/,1tume.l/,'b1tO Y1Ca3am'b:

1)	OAA 1te06XOOuMoao npU31ta1Ca -	lim		an;
		n-+oo
2)	OAA 1-ao U 2-ao npU31ta1C06 Cpa61te1tU.H. - 06Ut,uiJ. "Me1t pRoa,				C 'IC(JmOp'btM
Cpa61tU6aemC.H. Oa1t1t'btiJ. pRO;
3)	OAA npU31ta1Ca )1,MaM6epa -	lim	a~+1;
	n-+oo			n
4) OAA npU31ta1Ca KoU/,u - lim		'(,!'iln;			+00
	n-+oo
			nep6006pa31ty1O OAA f(x) U		Jf(x) dx.
5)	OAA U1tmeapM'b1tOaO npU31ta'ICa -
					a

B 3aOa"l,aX 1.3.7-1.3.14 OAA onpeOMe1tU.H. UHmepaa.tta CXOOUMocmu ucnO.!/,'b- 306am'b npU31ta1C )1,Ma.M6epa. B 3aOa"l,aX 1.3.15-1.3.20 OAA onpeOMe1tU.H. UHtnepaa.tta CXOOUMocmu UCnO.l/,'b306am'b npU31ta1C KoU/,u.

00 xn

1.3.7. E,· n=l n.

00

1.3.9.E n!xn.

n=l

00

1.3.11.E nxn.

n=l

00 xn

1.3.13. En' n=l

1.3.8.	00	(x _ 2)n-1
	E	(	)"
	n=l	n	+ I .

00

1.3.10.E (n + 2)!(x + I)n.

	n=l	(3 - x)2n
	00
1.3.12.	E	-"---=--
	n=l	.,fii
1.3.14.	00	n
	E	(_I)n-1_X_.
	n=l	n + I

00

00

1.3.15.E nnxn. 1.3.16. E nn+1(x - 3)n.

	n=l	xn		n=l	(x + 2)n+1
1.3.17.	00		1.3.18.	00
	En'			n~l	(n + I)n .
	n=l n
1.3.19.	f	(n; Ifx2n.	1.3.20. f		(n + I )n (x _ 2)2n+1.
	n=l			n=l	2n + I

38

a

aAA onpeae.l!eHUSI UHmepOa.l!a CXOaUMocmu ucno.ll'b-

1.3.21.	1.3.22.	00	(2x)n
		~	3~·
		n=l	yn
		00		(x _ 2)2n
1.3.23.	1.3.24.	~(_l)n			n
		n=l	(2 _ X)2n-2
1.3.25.	1.3.26.	00
		~		n
		n=l			n
		00
1.3.27.	1.3.28.	~	(_l)n_X_.
		n=l		n + 2
1.3.29.	1.3.30.	00	Xn(n-l)
		~	'2n		·
		n=l	n.
1.3.31.	1.3.32.	00	(x + 2)2n-l
		n~l		3n
1.3.33.	1.3.34.	00	(2n)!(x + 7)n+l
		n~l	(x -	3n - 1
1.3.35.	1.3.36.	00		3)n
		~ (		+	)'.
		n=l	2n		1 .
1.3.37.

HaiJ.mu 'lCpya CXOaUMocmu pSlaa. Y'lCa3am'b npUMe'HJleM'bte npU3Ha'ICU.

	00				00	(z + 2i)2n
1.3.38.	~ n!(z - i)n.			1.3.39.	~	n	2
	n=l	(z - 2i)n			n=l
1.3.40.	00			1.3.41.	f	(-n3.)n (z - i)n.
	~	n	n·
	n=l				n=l	n + z
	00	n
1.3.42.	~~.

n=l nz

AononHMTenbHble 3aAClHMR

HaiJ.mu o6.11acm'b CXOaUMocmu pSlaa. Y'lCa3am'b npUMe'HJleM'bte npU3Ha'ICU. ,D,o- nO.llHUme.ll'bHO Y'ICa3am'b:

1) aAA Heo6xoauMoao npU3Ha'ICa -		lim		an;
		n--+oo
2)	aAA 1-zo U 2-ao npU3Ha'ICOO cpaOHeHUSI - o6Ut,uiJ. "UIeH pSlaa,				c 'lCOmOp'btM
cpaOHuoaemc.n aaHH'btiJ. pSla;
3) aAA npU3Ha'ICa ,D,a.llaM6epa -		lim	a~+l;
	n--+oo			n
4)	aAA npU3Ha'ICa KOULU - lim	~;
	n--+oo		nepooo6pa3HY'lO aAA f (x) U		+00
5)	aAA UHmeZpa.ll'bHOZO npU3Ha'ICa -				f(x) dx.
					J

B 3aaa"tax 1.3.43-1.3.46

300am'b npU3Ha'IC ,D,a.I!a.M6epa.

39