Сборник задач по высшей математике 2 том

a) 1 + 2 + 3 + ...

+ n + ...

;

1

1

1

1

6) 1.2 + 2·3 + 3·4 + ...

+ n(n + 1) + ...

.

Q a) TaK KaK '1JIeHbIp.H.D:a 1 + 2 + 3 + ...

+ n + ...

npe.n.CTaBJI.HIOT co60ii

OTCIO.D:a lim Sn = lim

1

(n + n

2

)

= +00. CJIe.n.oBaTeJIb-

1 +2 n . n = lim -2

n-+oo

n-+oo

n-+oo

HO, p.H.D: pacXO.D:UTC.H. TaKUM 06pa3oM Sn =

1 +2 n . nj lim Sn = +00; PM

pacXOMTC.H.

n-+oo

111

6) TaK KaK

(

n

n+1

)

= n; - --1' TO

n+

1

1

1

1

=

Sn = 1.2 + 2·3 + 3·4 + ... + n(n + 1)

= (1 _ 1) + (1 _1) + (1 _1) + ... + (_1__ 1) + (1 __1_) =

2

2

3

3

4

n-1 n

n n+1

11111

1111

=1- 2 + 2

- a+ a-4+···+n-1-n;+n;-n+1 =

1

1

=1+0+0+ ... +0- n +1 =1- n

+1·

OTCIO.D:a

lim

Sn

=

lim

(1 - ~1) = 1. 3Ha'lUT, p.H.D: CXO.D:UTC.H,

U ero

n-+oo

n-+oo

n

+

CYMMa paBHa 1.

OKOH'IaTeJIbHO:Sn = 1 -

~1; lim

Sn = 1j PM CXOAUTC.H.

•

n +

n-+oo

,lI,JIJI. 'X:a:HCooeo pSioa 6

3aoa"tax 1.1.2-1.1. 8:

1) HanUCam'b rjJOPMY.II.Y "tacmu"tHoit cY.M.M1>I Sni

2) Haitmu

lim Sn U.II.U 00'X:a3am'b, "tmo amom npeoe.ll. He cyw,ecm6yemi

n-+oo

3) cOe.ll.am'b 61>1600 0 CxooUMocmu U.II.U pacxooUMocmu pSioa.

1.1.2.

1 -

1 + 1 -

1 + ... + (-1)n-l + ...

1.1.3.

1 + 3 + 5 + ... + (2n -1) + ...

1.1.4.

2 -

4 + 6 -

8 + ... + (_l)nH . 2n + ...

1.1.5.

1 + 2 + 4 + ... + 2n - 1 + ...

00

2

1

1.1.6.

La· 2n-1

n=l

1.1.9.

HaihH npe,ll;eJI npH n -+ 00 06m.ero qJIeHa pH,Il;a an. ECJIH

lim an "#

n---+oo

"# 0, TO, npHMeHHH Heo6xOMMbIii npH3HaK CXOMMOCTH, YCTaHo-

BHTb, qTO pH,Il; pacXOMTCH.

f

a) f

n + 1.

6)

n+2

n=l 2n + 1

n=l In(n + 1)

B) ~ +-.

00

2

n=l n + 2

a a) Haii,ll;eM npe,ll;eJI o6m.ero qJIeHa PMa:

1·

l'

n + 1

1man = 1m

- 21=

n---+oo

n---+oo

n

+

[Pa3,1l;eJIHM qHCJIHTeJIb H 3HaMeHaTeJIb ,Il;p06H Ha n]

lim

1+1

lim (l+k)

1

"# 0,

__n_ = n---+oo

(2 + k)

= -2

n---+oo

2 + k

lim

3HaqHT, pH,Il; paCXO,ll;HTCK.

n---+oo

IhaK, lim an =

1

-2 ; pH,Il; pacxoMTcH.

n---+oo

6) TaK KaK npH n -+ 00 HMeeM (n + 2) -+ 00 H In(n + 1) -+ 00, TO )l;JlH

HaxO>K,Il;eHHH npe,ll;eJIa lim

an BOCnOJIb3yeMcH npaBHJIOM JIonHTaJIH:

1

n~oo

lim

x + 2

= lim

(x + 2)'

= lim

~

= lim (x + 1) = 00.

x---+oo In(x + 1)

x---+oo (In(x + 1))'

x---+oo

ex + 1)

x---+oo

O

1·

an =

l'

n + 2

) =

..J.

0

,H pH,Il; paCXO,ll;HTCH.

TClO,Il;a CJIe,D;yeT, qTO

1m

1m

1 (

00 r

n---+oo

n---+oo

n n + 1

B) Haii)l;eM npe,ll;eJI 06m.ero qJIeHa pH,Il;a:

.

.

n 2

hm an = hm ---

n---+oo

n---+oo n3

+ 2

[Pa3,1l;eJIHM qHCJIHTeJIb H 3HaMeHaTeJIb Ha n 3 .]

.

n 2 : n 3

1

lim 1

0

- 0

•

n

n---+oo n

= hm

(n3 +2): n 3

= hm --- =

--- ;---.,- -

I

.

n---+oo

n---+oo 1 +.1..

lim (1 +

23) -

-

n 3

n---+oo ..

n

TaK KaK

lim an = 0, TO ,Il;aHHbIii pH,Il; MO>KeT CXOMTbCH, a

MO)KeT H pacxo-

)J,HTbCH.

n---+oo

B 3aoa"tax 1.1.10-1.1.17 Hai/,mu npeoe.n. npu

n -t

00

o6tqezo "t.lleHa

pSloa

an. Ec.IIu

lim

an "# 0, mo, npUMe'HJiJI Heo6xoOUM'bI,i/, npU3Ha1l: CXoouMocmu,

n-+oo

ycmaH06Um'b, "tmo PSlO pacxooumCSl.

1.1.10.

00

n + 2

1.1.11.

00

n2 + 1

L - '

L

(n + 2)

3'

n=l

2n -

3

n=l

00

n

00

n2 + 1

1.1.12.

5

1.1.13.

L -

L

arctg-+ ·

n=l

n

+ l'

n=l

n

3

1.1.14.

00

•

1

1.1.15.

00

(_I)n-l.n

L

sm n·

L

) .

n=l

n=l

In(n + 1

1.1.16.

00

1

00

(

3)n

n~l

(2 + (_I)n)n

1.1.17.

n~l

1 + n .

1.1.18.

IIpHMeHHH I-it npH3HaK cpaBHeHHH, HCCJIe)];OBaTb Ha CXO)l;HMOCTb

~ 2+sinn

PH)]; L."

n .

n=l

•

~ -1, TO 2 +

•

2+sinn

1

~1

Q TaK KaK smn

smn ~ 1, oTKy)];a

n

~ n' PH)]; L." n

n=l

~ 2+sinn

•

pacXO)];HTCH, 3HaqHT, pacXO)];HTCH H 60JIbWHit PH)]; L."

n .

n=l

1.1.19.

~ arctgn + 1

00

5n + 1

L."

n

2

•

1.1.20. L - 2n '

n=l

n=l

1

1.1.21.

n=l ..jii

1.1.22.

00

n~l

(n + I)!

f

Inn.

PH)];.

6) f

)

~

n+2

n..jii+ 2 .

aL . ,,2

•

n=l vn6 + 2n - 2

n=l n

+ n + 1

B)

00

1

n+l

L

3r,;: In -n-'

n=l

yn

n+2

~

HeOrpaHHqeHHO pacTyT

~ a) qHCJIHTeJIb H 3HaMeHaTeJIb )];P06H

2

T. e. qHCJIHTeJIb «pacTeT KaK n» npH n -t 00.

B

1·

n + 2

1

OJIee CTporo: 1m

- n - =

,

MeHaTeJIb «pacTeT KaK n

2

n -t 00. BOJIee CTpOro:

•

n2 + n + 1

= 1,

»npH

hm

2

n-+oo

n

qTO TaK:lKe MO:lKHO 3anHcaTb B BH)];e: n 2 + n + 1 '"n2 ,

n -t

00 (nOCJIe)];OBa-

TeJIbHOCTH n2 + n + 1 H n2 :3KBHBa.JIeHTHbI npH n -t 00).

,

(n + 2)

..... n

n

1

TaKHM 06pa30M, 2

+ n +

1) ..... n

2 ..... 2'

= n' B ,Il;pyrHx 0603HaqeHHHX:

(n

n

r

( n + 2 1)

r

n

2

+ 2n

=

n~

n2 + n

+ 1 : n

= n~~ n2

+ n + 1

lim

(n 2 + 2n) : n2

= lim

1 + ~

= 1.

n-+oo (n2 + n + 1) : n2

n-+oo 1 + 1 + ,1"

n

n2

00

1

TaK KaK pH,Il; ~ n pacXO,Il;HTCH, TO paCXO,Il;HTCH H HCXO)J;HbIii pH,Il;.

n=l

6) YqHTbIBaH, qTO H qHCJIHTeJIb H 3HaMeHaTeJIb ,Il;p06H

ny'n+ 2

vn6 +

He-

2n - 2

(n -t 00).

00

1

TaK KaK pH,Il;

~

3

CXO,Il;HTCH, TO CXO,Il;HTCH H

HCXO,Il;HbIii pH,Il;.

n=l n2

= In ( 1 + k)

..... k (n -t

Vn In n ~ 1

B) TaK KaK In n ~ 1

00), TO

-

1

1

1

(n -t

00). PH,Il;

00

1

. n = 4

~

4

CXO,Il;HTCH, 3HaqHT, CXO)J;HTCH H

vn

n3

n=l n3

HCXO)J;HbIii pH,Il;.

•

HCC.IIeiJo6am'b pS&iJ 'Ha cxoiJu.Mocm'b,

npU.Me'HJIJI 2-iJ. npU3'Ha'IC Cpa6'He'H'USl. Y'lCa-

3am'b o6w,uiJ. "t.lle'H ps&iJa, c 'lComop'bl..M Cpa6'HU6aemCSl

iJa'H'H'bI.iJ. ps&iJ.

1.1.24. ~-2-'

1.1.25.

I:

2-n

00

n

+

5

n=l n

-

2

n=l n3 +n-l

1.1.26.

I:

n 2 + 2

1.1.27.

00

1

n=l 3n + l'

n~l

-";=n=2=+=3

1.1.28.

I:

n -1

1.1.29.

I:

y'n +3 vn.

n=l Jn3 + 3n -

1

n=l

n+ W

1.1.30.

I: In (n2 ~1).

1.1.31.

~

·2

1

L.J

arcsm

. r,;:'

n=l

n

n=l

yn

1.1.32.

00

1.1.33.

I: n5 • tg3

22 ,

~ y'n'sin 11'2'

n=l

n

n=l

n

1.1.34.

00

2n + 3

~

-n 52'

n=l

+

00

n 5

00 nn

a)

E

n+!'

6)

E -,'

~

n=13

an +l

n=1 n,

\"I a) llpe06pa3yeM BbIpaJKeHHe

----a;;-:

an+!

=

(n + 1)5

: ~ = (n + 1)5 , 3n+!

= I, (1

1)5

an

3(n+!)+!

3n+!

n5

3n+2

3

+ n

TaK KaK it -t 0npH n -t 00, TO (1 + it)

-t 0H (1 + it)5 -t 0npH n -t 00,

3HaQHT,

'

an+! 1

l' (1

1) 5

1 1

l1m

-a-=-3

1m

+n

=-3<'

n-+oo

n

n-+oo

an+l

(n + l)n+!

nn

(n + l)n+!

n!

----a;;-

(n + I)!

:

n! =

nn

(n + I)!

=

= (n + l)n, (n + 1) ,

1,2,3, .. " n

=

(n + l)n =

(1 + 1)n ,

nn

1 ' 2 ' 3 ' , , , ' n ' (n + 1)

n

n

TO

lim

a~+!

= lim

(1 + it)n = e > 1 (2-ii 3aMeQaTeJIbHbIii npe)];eJI),

n-+oo

n

n-+oo

H, 3HaQHT, HCXO)];Hblii PH)]; pacXO)];HTCH,

•

' an+l

l1m -- ,

n-+oo an

00

2n

00

n 3

1.1.36.

E

2'

1.1.37.

E 3n'

n=1 n

n=1

00

3n

00

(n!)2

1.1.38.

E "

1.1.39.

E - ()"

n=1 n,

n=1

2n,

00

nn

E1, 4 ' .. , ' (3n -

2)

1.1.40.

E

'2n '

1.1.41.

n=1

n!2n

'

n=1 n,

1.1.42.

E 1,3,5, .. " (2n - 1)

n=1 2,7, 12, .. , ' (5n -

3)'

1.1.43.

IIccJIe)];oBaTb PH)];bI Ha CXO)];HMOCTl>, npHMeHHH npH3HaK KomH:

a)

E(n + 2 )3n+!,

6)

En' (1 _ it)n2

n=1 2n + 1

n=1

a a) YqHTbIBaJI, qTO

3n+ 1

1

~=n ( n+2 )3n+l = (n+2 ) -n - = (n+2 )3+n

'

2n + 1

2n + 1

2n + 1

1·

n + 2

=

1

1·

(3

+

1 )

=

3

, nOJIY'IHM

a 1m

- 21

-2 H

1m

11

n---+oo n +

n---+oo

.

.

( n + 2 ) 3+ ~

( 1) 3

1

< 1.

J~~ va;;

= J~~

2n

+ 1

="2

= 8

I1CXO,LI,HbIA PM CXO,ll;HTCfI no npH3HaKY KOWH.

n 2

'l /

(

1 ) n

2

1

(

1 -

1 ) -

1

(

1 ) n

'

6) TaK KaK va;; = Vn·

1 - 11

= nn.

11

n

= nn.

1 - 11

TO OCTaeTCfI HaATH npe,ll;eJIbI

lim

n~ H

lim

(1 - k)n.

n---+oo

n---+oo

1

1.

1

1) llOCKOJIbKY n n = e ln (n n ),

r,ll;e In(n n) = kIn n, TO no npaBHJIy .ITonH-

TaJIfI

.

Inn

.

(Inn)'

.

k

= 0,

hm

rt

= hm

-(-),- =

hm -1

n---+oo

n---+oo

n

n---+oo

1

= lim

1

= 1.

OTKY,ll;a lim nn

enlnn = eO

n---+oo

n---+oo

2) TaK KaK

lim (1 + ~)n

= eO< (CJIe,ll;CTBHe H3 2-ro 3aMe'laTeJIbHOrOnpe-

n---+oo

,ll;eJIa), TO lim

(1- k)n = e- 1. OTCIO,ll;a

n---+oo

1·

'l~

1·

1.

(1

-

l)n

=

1·

1.

1·

(1

-

l)n

= e-

1

1

1m

'{ran =

1m nn .

-

1m nn.

1m

-

= - < 1,

n---+oo

n---+oo

n

n---+oo

n---+oo

n

e

H, 3Ha'lHT,HCXO,ll;HbIA pfl,ll; CXO,ll;HTCfI.

•

lim va;.

n---+oo

f (n -1 )n

1.1.44.

00

n 2

1.1.45.

n~l(l+k)

n=l

2n + 1 .

n

E(n - 1 )n(n-1).

1.1.46.

E(2n-l)2

1.1.47.

n=l

3n + 1 .

n=l

n + 1

1.1.48.

E(arCSin kr·

1.1.49.

00

(3n + 2)n

Ln· --

n=l

2n + 1 .

n=l

1

1.1.50.

00

l1cCJIe,ll;OBaTb Ha CXO,ll;HMOCTb PM L

-1-'npHMeHfIfI HHTerpaJIb-

n=2 n nn

)l;JIfI <PYHKIJ;HH f(x)

HbIA

npH3HaK. YKa3aTb

nepBo06pa3HYIO

+00

H ! f(x)dx.

a

= -11 . llpOBepHM npHMeHHMOCTb HH-

1

f(x)

Q TaK KaK an = -1- , TO

n

nn

x nx

IIycTb 2 < Xl < X2. Tor,ll;a In Xl

< In X2 II Xl In Xl < X2 In X2, oTKY,ll;a

f(xt} =

1

>

1

= f(X2).

Xl

1

X2

1

nXI

nX2

IIepBoo6pa3Hoit MH <PYHKIJ;IIII f(x)

HBJIHeTCH, HanpllMep, <PYHKIJ;IIH Inlnx.

+00

BbIQIlCJIHH Heco6cTBeHHbIit IIHTerpan

f,

nOJIYQIlM

+00

M

2

~ =

lim

f~ =

lim

(InlnxIM) =

f xlnx

M-++oo

xlnx

M-++oo

2

2

2

=

lim (In In M -lnln2) = +00.

M-++oo

+00

TaK KaK Heco6cTBeHHbIit IIHTerpan

f

dlx

paCXOMTCH, TO paCXOMTCH II

00

1

X

nx

pH,ll; L -- .

2

•

n=2 nlnn

HCC.IIe006am'b PSlO Ha CXOO'UMOcm'b,

np'UMeHJiJI 'I.mmeepa.ll'bH'b/,fJ. npU3Ha'IC. Y'lCa-

+00

3am'b nep6006pa3HY'KJ OM tjjYH'IC'qUU f(x)

U f

f(x) dx.

a

1.1.51.

00

1

1.1.52.

00

1

L -

n~l (n + 1) In(n + 1)

n=2 nv'ln n'

1

1.1.53.

00

1.1.54.

00

1

L

2

+ 1)

•

L

p'P> O.

n=l

(n + 1) In (n

n=l

n

HCC.IIe006am'b PSlO Ha CXoOUMOCm'b. Y'lCa3am'b

npUMeHSleM'b/,e npU3Ha'ICu. ,40-

nO.llHUme.ll'bHO Y'lCa3am'b:

1) OM He06xoo'UMoeo npU3Ha'ICa -

lim

an;

n-+oo

C 'lCOmOp'b/,M

2) OM l-eo U 2-eo npU3Ha'IC06 cpa6HeH'USl -

06~ufJ. "MeH pSloa,

cpa6HU6aemCSl OaHH'b/,fJ. PSlO;

9) OM npU3Ha'ICa ,4a.llaM6epa -

lim

n

aa+1;

4) OM npU3Ha'ICa Kowu -

n-+oo

n

lim

~;

00

n-+oo

5) OM UHmeepa.ll'bHOeO npU3Ha'ICa -

nep6006pa3HY'KJ OM f(x) U

ff(x) dx.

E2n+3.

a

1.1.55.

1.1.56.

00

n 2

L 3n'

n=l 3n - 2

n=l

1.1.73.

~ (_I)n, (2n - 1) = -1 + 3 - 5 + 7 - .. , + (_I)n , (2n - 1) + .. ,

n=l

1

1

1

1

1.1.74.

oo

~

--1 =1+- + - +"'+--1+'"

n=15n-

5

52

5n-

1.1.75.

n~l

2'+ (_I)n '2'= 2 + 3 + 2 + 3 + .. , + 2 + 3 + .. ,

00

(5

1)

1.1.76.

E 2n + 1

3 _ + _7_ + + 2n + 1 +

n=l n 2(n + 1)2 - 12 , 22

32 ,42

.. ,

n 2(n + 1)2

.. ,

Ha11mu

npeiJeJ& o6w,eeo

"t.l&eHa

pSl.iJa

an' Ec.l&u

lim

an "# 0, mo,

npU.M.e'H.SI.SI.

n-+oo

Heo6xoiJu.M.'bI.11 npu3Ha-,. cxoiJU.M.ocmu,

ycmaH06Um'b,

"tmo PSl.j) pacxoiJumCSl.,

1.1.77.

00

Eln 3n - 1,

~_n_,

1.1.78.

n=l 3n-l

n=l

2n + 3

1.1.79.

00

1

1.1.80.

00

n+l

~ cos 2"

~ arcctg - 2 -- '

n=l

n

n=l

n - 3

1.1.81.

00

(_I)n+1

1.1.82.

00

2n

n~l

n+ViO

~

3n '

n=l

1.1.83.

00

n+3

1.1.84. E

2

-

1

•

n=13n

1.1.85.

f: cos ~k) .

1.1.86. n~ln \n.

n=l

00 2n - 1

1.1.87.

~ In(nn+ 1).

1.1.88.

L.J

E - 5l'

n=l

n

+

n=l

00 1 (

1 )n2

1.1.108.

~ (n+ 1 )n-l.

1.1.107. n~l 2n

1 + n

L.J

3

1

n=l

n-

1.1.109.

1.1.110.

00

1

n~l lnn(n + 1)

1.1.111.

1.1.112.

00

1 (2n l)n

E --

n=l n 3n + 1 .

+00

3am'b nep6006pa3HY'IO a.llSl. tj}YH'lCq'U'U

f(x) 'U ! f(x) dx.

a

1.1.113.

~ Inn

1.1.114.

00

1

n~2 (2n + 1) In(2n + 1)

n=2

n·

L..J

00

1

00

1

1.1.115.

n~2 -n-:-In-n~ln--:-In-n-

1.1.116. n~2

-n-In-n-("":;ln'--ln-n-)-2

1)

a.llSl. He06xoa'UM020 np'U3Ha'ICa -

lim

an;

n-+oo

2)

a.llSl. 1-20 'U2-20 np'U3Ha'IC06 cpa6HeH'USI. - 06w,'UiJ. "t.lleH pSl.aa,

C 'lComop'bI..M

cpa6H'U6aemCSl. aaHH'bI.iJ. pSl.a;

3) a.llSl. np'U3Ha'ICa j(a.llaM6epa -

lim

a~+1;

n-+oo

n