246503

4

~ISLOWYE RQDY

STI^NYE SUMMY. fORMALXNOE WYRAVENIE WIDA a1 +a2 + : : :+an +: : : =

1 an, GDE

f

an

1

{ BESKONE^NYJ NABOR (NE OBQZATELXNO RAZNYH) ^ISEL,

nP=1

gn=1

NAZYWAETSQ ^ISLOWYM RQDOM (ILI PROSTO RQDOM) S OB]IM ^LENOM an.

(rASSMATRIWA@TSQ TAKVE RQDY

1 an, W KOTORYH NUMERACIQ NA^INAET-

P

n

n=s

SQ NE S 1,

A S DRUGOGO CELOGO ^ISLA s.) sUMMA

P

ak PERWYH n ^LENOW

k=1

RQDA NAZYWAETSQ n-J ^ASTI^NOJ SUMMOJ DANNOGO RQDA I OBOZNA^AETSQ

^EREZ Sn. rQD

1 an NAZYWAETSQ SHODQ]IMSQ, ESLI SU]ESTWUET KONE^NYJ

P

n=1

PREDEL S = lim

Sn POSLEDOWATELXNOSTI Sn ^ASTI^NYH SUMM \TOGO RQDA.

n!1

1 an =

S I GOWORQT, ^TO RQD

1 an SHODITSQ

w \TOM SLU^AE PI[UT

P

P

n=1

n=1

(K ^ISLU S),

GDE ^ISLO S NAZYWAETSQ SUMMOJ RQDA

1 an. qSNO, ^TO

P

n=1

S TAKVE RAWNO

lim Sn

;

1. rQD

1 an NAZYWAETSQ RASHODQ]IMSQ, ESLI

PREDEL lim Sn

n!1

nP=1

NE SU]ESTWUET ILI BESKONE^EN (W \TOM SLU^AE TAKVE

1.1.2. aBSOL@TNO I USLOWNO SHODQ]IESQ, POLOVITELXNYE I

NEOTRICATELXNYE RQDY. rQD

1 an NAZYWAETSQ ABSOL@TNO SHODQ]-

P

n=1

IMSQ,

1

I

1

ESLI OBA RQDA nP=1 janj

nP=1 an SHODQTSQ, PRI^EM W 1.2.5 MY

POZVE DOKAVEM, ^TO DOSTATO^NO TREBOWATX SHODIMOSTI TOLXKO RQDA

1

1

SLEDUET SHODIMOSTX RQDA

nP=1 janj, T.E., IZ SHODIMOSTI

RQDA nP=1 janj

1 an. rQD

1 an NAZYWAETSQ USLOWNO SHODQ]IMSQ, ESLI ON SAM SHODIT-

P

P

n=1

n=1

SQ, A RQD

1

1

nP=1 janj RASHODITSQ.

rQD nP=1 an

= a1 + a2 + : : : + an + : : :

NAZYWAETSQ NEOTRICATELXNYM (POLOVITELXNYM), ESLI WSE EGO ^LENY

NEOTRICATELXNY (POLOVITELXNY), T.E. an

> 0 (an > 0) DLQ WSEH

n 2

N. rQD S ^LENAMI PROIZWOLXNYH

ZNAKOW TAKVE NAZYWAETSQ

ZNAKOPEREMENNYM RQDOM.

oB]IE SWOJSTWA ^ISLOWYH RQDOW

5

1.1.3. pRIMER. rQD

1

1

=

1

1

SHODITSQ I EGO SUMMA

n(n + 1)

(k

;

1)k

n=1

k=2

RAWNA 1.

X

X

1

1

1

tAK KAK an =

= n

;

, TO

n(n + 1)

n + 1

1

1

1

1

1

1

Sn = a1 + a2 + : : : + an = 1 ;

2 +

2 ; 3

+ : : : + n ;

= 1 ;

:

n + 1

n + 1

tAK KAK

lim Sn = lim

1

1

= 1,

TO RQD

SHODITSQ

I

; n + 1!

1

1

n!1

n!1

= 1.

n=1

n(n + 1)

X

eSLI RQDY 1 an I

1 bn SHODQTSQ K

1.1.4. dEJSTWIQ NAD RQDAMI.

n=1

n=1

P

P

^ISLAM A I B SOOTWETSTWENNO, TO DLQ L@BYH ^ISEL I

RQD

1

( an + bn) SHODITSQ I EGO SUMMA RAWNA A + B

n=1

P

+ bn) = n1=1 an + n1=1 bn.

T.E. n1=1( an

P

P

P

1 an,

1 bn

I

/ pUSTX An, Bn I Sn

{ n-YE ^ASTI^NYE SUMMY RQDOW

n=1

n=1

1

( an + bn) SOOTWETSTWENNO. qSNO, ^TO

P

P

Sn = An + Bn. iZ SWOJSTW

n=1

SLEDUET, ^TO

PREDELOWP

lim Sn = lim ( An + Bn) = lim An

+

lim Bn = A + B: .

n!1

n!1

n!1

n!1

TO EGO OB]IJ ^LEN STREMITSQ K NUL@ PRI n

! 1. pO\TOMU ESLI

lim

an = 0, TO RQD

1 an RASHODITSQ.

n!1

6

n=1

/

eSLI

RQD

P n1=1 an

SHODITSQ K

^ISLU

S, TO

lim an = lim

Sn

lim

P

1 = S

S = 0.

.

Sn

;

n!1

n!1

; n!1

;

1.1.6. pRIMER RASHODQ]EGOSQ RQDA SO STREMQ]IMSQ K NUL@

OB]IM ^LENOM. 1

/ qSNO, ^TO nlim!1 p

= 0, PRI^EM PRI n 2

n

1

1

1

1

1

1

n

= pn n;! +1.

Sn = p

+ p

+ : : : + p

>

p

+ p

+ : : : + p

= p

1

2

n

n

n

n

n

1 an RASHODITSQ. .

!1

pO\TOMU lim

Sn = +

1

I RQD

!1

P

n

n=1

6

~ISLOWYE RQDY

1.1.7. bESKONE^NAQ GEOMETRI^ESKAQ PROGRESSIQ. rQD n1=1 xn;1 RA-

1

P

SHODITSQ PRI

jxj 1 I SHODITSQ K

PRI

jxj < 1. pO\TOMU DLQ

1 ; x

L@BOGO

NENULEWOGO ^ISLA

BESKONE^NAQ

GEOMETRI^ESKAQ PROGRESSIQ

+ x + x2 + x3 + : : : RASHODITSQ PRI jxj 1 I SHODITSQ K

1 ; x

PRI jxj

< 1.

lim lim xn;1

/ tAK

KAK

= 0

PRI

j

x

j

1,

TO

PRI

j

x

j

1

RQD

n1=1 xn;1

n!1

6

PRIZNAKU

RASHODITSQ PO

NEOBHODIMOMU

SHODIMO-

STI

P

dOPUSTIM

TEPERX,

^TO

x

<

1.

tAK

KAK

1.1.5.

Sn = 1 + x + x2 + : : : + xn;1 =

1

; xn

=

1 j j

xn

,

TO PRI

x

<

1

1 ; x

; 1 ; x

IMEEM

lim Sn

=

1

. .

1 ; x

j j

; x

1.1.8.

n!1

1

SUMMY SHODQ]EGOSQ

RQDA

OGRANI^ENY

W

wSE ^ASTI^NYE

/ pUSTX RQD

1 an SHODITSQ K ^ISLU S, T.E.

lim Sn = S. tOGDA DLQ

WSEH n 2

N

nP=1

n!1

NA^INAQ S NEKOTOROGO NOMERA k, WSE ^ASTI^NYE SUMMY Sn

OTLI^A@TSQ OT ^ISLA S MENX[E ^EM NA 1. kROME TOGO, SU]ESTWUET TAKOE

MENQET SHODIMOSTX ILI RASHODIMOSTX RQDA, TO RQDY

1

bn,

1 bn I

n=1

n=1

1

P

P

bn LIBO WSE SHODQTSQ, LIBO WSE RASHODQTSQ. eSLI WSE \TI TRI

nP=1 3

RQDA SHODQTSQ,

TO IZ PERWOGO PRIZNAKA SRAWNENIQ 1.2.1

I NERAWENSTW

0 < an < 3

1

bn SLEDUET SHODIMOSTX RQDA nP=1 an. eSLI VE WSE \TI TRI

RQDA RASHODQTSQ, TO IZ PERWOGO PRIZNAKA SRAWNENIQ 1.2.1 I NERAWENSTW

0 <

bn < an SLEDUET RASHODIMOSTX RQDA

1 an. .

n=1

P

1

I

1

1.2.3.

tRETIJ

PRIZNAK

SRAWNENIQ

pUSTX

an

bn

{

.

n=1

n=1

an+1 6 bn+1

P

P

POLOVITELXNYE RQDY I

DLQ WSEH NOMEROW

n NA^INAQ S

NEKOTOROGO NOMERA.

an

bn

(1)

eSLI RQD

1 bn

SHODITSQ, TO RQD

1 an TOVE SHODITSQ.

n=1

n=1

P

P

(2)

eSLI RQD

1 an

RASHODITSQ, TO RQD

1 bn TOVE RASHODITSQ.

n=1

n=1

P

P

/ pO USLOWI@ a2

6 b2

a3

6 b3

: : :

an

6

bn

.

a

a

1

b

b

2

a

b

1

2

n;1

n;1

pEREMNOVAQ OTDELXNO WSE LEWYE I WSE PRAWYE ^ASTI \TIH NERAWENSTW,

an

bn

a1

a1

POLU^IM a1 6 b1

, OTKUDA an 6 b1 bn.

tOGDA an

6 bn,

GDE

=

b1 ,

PRI^EM RQDY

1 bn

I

1

bn LIBO OBA SHODQTSQ,

LIBO OBA RASHODQTSQ.

n=1

n=1

P

P

tEPERX PRIMENIM PERWYJ PRIZNAK SRAWNENIQ 1.2.1. .

1.2.4. pUSTX

1 an I

1 bn { POLOVITELXNYE RQDY.

n=1

n=1

P

P

1)

eSLI lim an = 0 I RQD

1 bn SHODITSQ, TO I RQD 1 an SHODITSQ.

n

!1 bn

n=1

n=1

P

P

2)

eSLI lim

an = +

1

I RQD

1 bn

RASHODITSQ, TO I RQD 1 an

RA-

n!1 bn

n=1

n=1

SHODITSQ.

P

P

/ 1). tAK KAK

lim

an = 0,

TO 0

< an

< bn DLQ WSEH n NA^INAQ S

n!1 bn

1 bn

NEKOTOROGO NOMERA.

kROME TOGO,

PO USLOWI@ RQD

SHODITSQ.

pO

n=1

PERWOMU PRIZNAKU SRAWNENIQ 1.2.1 RQD

P

1 an TOVE SHODITSQ.

n=1

P

pRIZNAKI SRAWNENIQ I INTEGRALXNYJ PRIZNAK

9

2). tAK KAK

lim

an = +

1

,

TO 0

< bn < an DLQ WSEH n NA^INAQ S

n!1 bn

1 bn RASHODITSQ. pO

NEKOTOROGO NOMERA. kROME TOGO, PO USLOWI@ RQD

n=1

PERWOMU PRIZNAKU SRAWNENIQ 1.2.1

RQD

P

1 an TOVE RASHODITSQ. .

n=1

P

1.2.5. eSLI RQD IZ MODULEJ ja1j + ja2j + ja3j + : : : SHODITSQ, TO RQD a1 +

a2 + a3 + : : : TOVE SHODITSQ.

janj SLEDUET,

/ iZ NERAWENSTW

;janj

6

an

6

^TO

0

6

an +

an

6 2 an . iZ

USLOWIQ WYTEKAET

SHODIMOSTX RQDA

1

2 an . pO

j

j

j

j

n=1

j

j

P

PERWOMU PRIZNAKU SRAWNENIQ 1.2.1 RQD 1 (an +

an ) SHODITSQ. tAK

n=1

j

j

1

1

1

P

1

KAK n=1 an

= n=1(an + janj ; janj) = n=1(an + janj) ; n=1 janj, TO RQD n1=1 an

X

X

X

X

P

SHODITSQ,

KAK RAZNOSTX DWUH SHODQ]IHSQ RQDOW. .

FUNKCIQ f(x) NEPRERYWNA, NEOTRICATELXNA I UBYWAET. tOGDA RQD

1 f(n) = f(1) + f(2) + : : : + f(n) + : : : I NESOBSTWENNYJ INTEGRAL

n=1

P

Z1f(x) dx LIBO OBA SHODQTSQ, LIBO OBA RASHODQTSQ.

1

/ tAK KAK f(x) UBYWAET, TO f(k + 1) 6 f(x) 6 f(k) DLQ WSEH x 2 [k k+1].

pO\TOMU f(k + 1) 6 kZ+1f(x)dx 6 f(k). pODSTAWLQQ W \TI NERAWENSTWA

k

k = 1 2 : : : n ; 1, POLU^IM

f(2) 6 Z2 f(x)dx 6 f(1) f(3)

6 Z3 f(x)dx 6 f(2)

: : :

1

2

f(n) 6 Zn f(x)dx 6 f(n ; 1):

n;1

n

n

n;1

6 Z

sKLADYWAQ \TI NERAWENSTWA, POLU^IM k=2 f(n)

f(x)dx 6 k=1 f(n).

pO\TOMU Sn ; a1 6 Zn f(x) 6 Sn;1

X

1

X

( ) GDE

Sn

{ n-Q

^ASTI^NAQ

1

SUMMA RQDA 1 f(n). iNTEGRALXNYJ PRIZNAK 1.2.6

SLEDUET TEPERX IZ

n=1

P

PRIWEDENNYH NIVE UTWERVDENIJ a) I b).