Сборник задач по высшей математике 2 том

KOHTponbHble BOnpOCbl M 60nee CnO)KHble 3aAaHMH

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

20

							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

§ 2. 3HAKOnEPEMEHHbiE PSlAbl

3HaKOyelJeAYtO~eCfl PfiAbI

~3'Ha'ICo"l.epeiJYI014'UMCJj H8.3hIBaeTCH PH)];, B KOTOPOM JII06hIe )J;Ba COCe)];HHX 'IJIeHa

HMeIOT pa3HhIe 3HaKH. TaKHM 06pa30M, 3HaKO'lepe,D;JIOIIIHii:cHPH)]; - 3TO PH)]; BH)];a

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, ... ).	$

IIycTb )];aH 3HaKO'lepe,D;JIOIIIHil:cHPH)]; (BH)];a (2.1) HJIH (2.2)). ECJIH BhInOJIHeHhI

)J;Ba YCJIOBHH:

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

TO PH)]; CXO)];HTCH.

~PH)];, CO)];ep:JKaIIIHiI: H nOJIO:JKHTeJIbHhIe H OTpHn;aTeJIbHhIe 'lJIeHhI,Ha3hIBaeTCH

3'Ha'ICOnepeMe'H'H'btM. B 'IaCTHOCTH,BCHKHiI: 3HaKO'lepe,D;JIOIIIHil:cHPH)]; HBJIHeTCH 3Ha-

KonepeMeHHhIM.

$

21

	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

M02KHO npHMeHHTb Bce npH3HaKH, HCnOJIb3yeMble npH Hcc.rre,n;OBaHHH PH,n;OB C noJI02KHTMbHblMH 'IJIeHaMH.

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

00

>r,;: ,D;JIH Bcex n. PM E .r,;: paCXO,D;HTCH, TaK KaK pacXO,D;HTCH pH,D;11

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

22

00	1
3HaKY cpaBHeHHH pacXO,D;HTCH H pH,D; E	.jii
n=12	n - l

IITaK, HCXO)1,HbIA pH,D; He HBJIHeTCH a6COJIIOTHO CXO)J;HIII.HMCH.

2. BbIHCHHM, CXO,D;HTCH JIH ,D;aHHblA 3HaKOqepe,D;yIOIII.HAcH pH,D;, npHMeHHH npH3HaK JIeA6HH:u;a.

a) IIpoBepHM, BbInOJIHHeTCH JIH HepaBeHcTBo an > an+! ,D;JIH a6COJIIOTHbIX BeJIHqHH qJIeHOB ,D;aHHOrO pH,D;a:

an =	1	1 >	1	=an +l·
	2.jii -		2v'n + 1 - 1

,I1;aHHoe HepaseHcTBo 3KBHBaJIeHTHO HepaseHcTBY 2.jii -1 < 2v'n + 1-1,

KOTopoe BepHo ,D;JIH JII060ro n = 1, 2, ••• 3HaqHT, an > an+! ,D;JIH Bcex HOMepoB

n= 1,2, ...

6)HaA,D;eM npe,D;eJI 06III.ero qJIeHa pH,D;a:

1·	1·	1	0
1man =	1m		=.
n-+oo	n-+oo 2.jii		+ 1

TaKHM 06PMOM, ,D;JIH ,D;aHHoro 3HaKOqepe,D;yIOIII.erOCH pH,D;a BbInOJIHeHbI

06a YCJIOBHH, CO,D;ep)KaIII.HeCH B npH3HaKe JIeA6HH:u;a, OTKY,D;a CJIe,D;yeT, 'ITO HCXO,D;HbIA PH)J; CXO,D;HTCH. O,D;HaKO OH He HBJIHeTCH a6COJIIOTHO CXO)J;HIII.HMCH,

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

00

CJIe,D;OBaTeJIbHO, 3HaKOnOCTOHHHbIA pH,D; E an pacxO,D;HTCH, a 3HaqHT, HC-

n=l

00

XO,D;HbIii pH,D; E (-I)na n He HBJIHeTCH a6COJIIOTHO CXO)J;HIII.HMCH. n=l

2. Tenepb BbIHCHHM, HBJIHeTCH JIH ,D;aHHbIii 3HaKOnepeMeHHbIii pH,D; cxoMIII.HMCH, HCnOJIb3YH npH3HaK JIeii6HH:u;a.

a) IIpoBepHM, BbInOJIHHeTCH JIH HepaBeHcTBo an > a n +l MH Bcex HOMePOB n, HaqHHaH C HeKOToporo:

an = 1 >	1	=an +l.
2n -Inn	2(n + 1) -In(n + 1)

23

3anHweM nOCJIe,n:OBaTeJIbHOCTb HepaBeHCTB, 9KBHBarreHTHbIX ,n:aHHOMY:

2n -Inn < 2(n + 1) -In(n + 1);

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

In n ~ 1 < 2;

In (1 + k) <: 2.

TaK KaK l+k ~ 2 < e, TO In (1 + k) < Ine = 1 < 2 .D:JIHJIIo6oron = 1,2, ...

3Ha'lHT,HepaBeHCTBO an > an+l BbInOJIHHeTCH.D:JIH BCex n = 1,2, ...

6) Hail:,n:eM npe.u,eJI o6111ero '1JIeHaPMa:

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

o6a YCJIOBHH, co,n:ep:>KalllHecH B npH3HaKe JIeil:6HHIJ;a, 3Ha'lHT,9TOT pH,n: cxo- ,n:HTCH. 1I3 9Toro H H3 Toro, 'ITOPH,n: He HBJIHeTCH a6COJIIOTHO CXO.D:HIII,HMCH,

OKOH'IaTeJIbHOCJIe.n:yeT, 'ITOpH,n: CXO,n:HTCH YCJIOBHO. •

00

1.2.3.IIccJIe,n:oBaTb Ha CXO,n:HMOCTb PM n~l(_I)n-l ; .

00

<) IIccJIe.n:yeM Ha CXO,n:HMOCTb PH,n: 2: an H3 a6COJIIOTHbIX BeJIH'IHH'1JIeHOB n=l

,n:aHHoro pH,n:a, T. e. PM:

00

L;'

n=l

HCnOJIb3YH npH3HaK ,I1;arraM6epa. )];JIH 9Toro CHa'iarranpeo6pMyeM BbIpa:>Ke-

an+l
HHe ---a;;-:
an+! = n + 1 : ..!!.. = n + 1 . ~ = (1 + 1) .1.
an 3n+! 3n	n	3n+!	n 3

Hail:.n;eM npe,n:eJI 9Toro Bblpa:>KeHHH:

1·	an+l
1m	-a-=
n-too	n

l' (	1)	1 1
1m	l+n	'-3=-3<1.
n-too

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.	•

1

1.2.4.IIccJIe,n:oBaTb Ha CXO,n:HMOCTb PM 2: (-I)n sin 2"00

n=l n

<) PacCMoTPHM PH,n: EIsin ~IH3 MO.n:yJIeil: '1JIeHOB,n:aHHoro pH,n:a, T. e. (TaK n=l n

KaK 0 < ~ < 1, H CJIe,n:OBaTeJIbHO, sin ~ > 0 ,n:JIH Bcex n = 1,2, ... ):

n	n
00	1
", .
L..J SIll 2"
n=l	n

24

BOCnOJIb3yeMcH 2-M npH3HaKOM CpaBHeHHH, ,lI,JIH '"IeroCpaBHHM 9TOT PM C

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,

n=l

T. e. pH,ll;:

~ 1 . 4 . 7 ..... (3n - 2) .

~ 3·5·7· ... · (2n + 1)

,I1;JIH OTBeTa Ha Bonpoc 0 CXO,ll;HMOCTH nOJIY'"IeHHOrOPMa npHMeHHM npH3HaK ,naJIaM6epa:

an +1 1 ·4· 7· .... (3n - 2)(3(n + 1) - 2) 1· 4· 7· .... (3n - 2) 3n + 1 ~ = 3·5·7· .... (2n + 1)(2(n + 1) + 1) : 3·5·7· .... (2n + 1) = 2n + 3'

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

CTaJIO 6bITb (TaK KaK He BbIllOJIHHeTCH He06xo,ll;HMblii npH3HaK CXO,ll;HMOCTH),

	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.	•

25

,l(0'ICa3am'b, "tmo PSliJ cxoiJumCSl

1.2.7.	00	(_1)n-1
	n~l '''':'In-:-(n.....:.-+-1~)
	00	(-l)n
1.2.9.	L:	.

n=2 n In nv'ln In n

,l(0'ICa3am'b, "tmo PSliJ cxoiJumCSl

YC.ll.06'H.0:

E (-1)n(2n+1)

1.2.8.

n=l n(n + 2) .

1.2.10.E(-l)n+1 1

n=l

2n - Vii

a6co.ll.tOm'H.0:

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)

00

1.2.15.L:(-l)n·n.

	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

HCC.II.eiJ06q,m'b PSliJ'bt 'H.a cxoiJUMoCm'b. Y'lCa3am'b npUMe'H.SIeM'bte npU3'H.a'ICu. ,l(o- nO.ll.'H.ume.ll.'b'H.O Y'ICa3am'b:

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

26

00	(3 + i)n
1.2.28. I1ccJIe,n:oBaTb Ha CXO,n:HMOCTb PM n~l	n. 2n

00

Q TIpHMeHHM K pH)JJ' 2: lanl H3 a6COJIIOTHbIX BeJIHqHH qJIeHOB ,n:aHHOI'O

n=l

pMa npH3HaK ,1l;anaM6epa:

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

00

Q TIpHMeHHM K pH)JJ' 2: lanl H3 a6cOJIIOTHbIX BeJIHqHH qJIeHOB ,n:aHHOI'O

n=l

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:

n=l yn

27

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

TaK KaK ,D;06aBJIeHHe (H y,D;aJIeHHe) npOH3BOJIbHoro qHCJIa qJIeHOB pH,D;a, pasHbIX HyJIIO, He BJIHHeT Ha ero CXO,D;HMOCTb, nOJIyqHM ,D;Ba pH,D;a:

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-

,D;HTCH, IIOCKOJIbKY B nyHKTe 1 3a,D;aQH yCTaHOBJIeHO, QTO HCXO,D;HbIii pH,D; He

HBJIHeTCH a6COJIIOTHO CXO,lJ;jJIII.HMCH, 3HaQHT, OH CXO,lJ;HTCH YCJIOBHO, •

28

lfcc.n.ea06am'b pSla'bl. Ha CXOaU.M.OCm'b. Y7&a3am'b npU.M.eHSle.M.'bI.e npU3Ha7&U. ,[(0- nO.n.HUme.n.'bHO Y7&a3am'b:

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

1.2.45.E (_I)n-1Cos~n.

n=l 2

00

1.2.46.~ (-I)nln(n + 1).

n=l	(n + 1)n(n-l}
00		.
1.2.48. ~(-I)n	-
n=l	n-I

00I

1.2.47.~ (_I)n-l n·2 •

n=l 2n

1.2.49.E (-I)nln n+3 .

+1n=l 2n

29