Сборник задач по высшей математике 2 том

B 3aoa"tax 1.3.47-1.3.49 ONi onpeoe.n.eHUSI l.mmepaaJl,a CXoouMocmu UCnOJl,b- 30aamb npU3Ha?i: KoU/,u.

1.3.43.	00	(x - 2)n			1.3.44.	00	(3x)5n
	~	n 2				~-.
	n=l					n=l	2n - 1
1.3.45.	00 n!xn+1				1.3.46.	00 3n1x n+2
	~	2n - 1 .				~	,.
	n=l	2n(2x + 3)n-l				n=l	n.
1.3.47.	00				1.3.48.	00	nn(x + l)n+l
	~	n	n			fl	3 n - 1
	n=l
	00	n2
1.3.49.	~~.
	n=l	n
KOHTponbHble BonpOCbI lit 60nee CnO)l(Hbie 3aWlHiltH
1.3.50.	MmKeT JIH HHTepBa.JI CXO.n;HMOCTH pH.n;a ~ anxn 6bITb TaImM:
	a)	(-2; 0);			6) (0; 2);
	B) (-3; 1);				r)	(-00; 00);
	~) (-3; 3).
1.3.51.					00
	l:I3BecTHo, qTO pH.n;				~ an(x - 3)n B TOqKe x = 2 pacXO.n;HTCH. "tho
					n=l
	MO:>KHO CKa3aTb 0 CXO,n;HMOCTH PMa B TOqKe:
	a) x = 5;				6) x = 3,5;
	B) x = 4.
					00
1.3.52.	l:I3BecTHo, qTO PM				~ an(x - 3)n B TOqKe x = 2 CXO.n;HTCH a6co-
					n=l
	JIIOTHO. qTO MO:>KHO CKa3aTb 0 CXO,n;HMOCTH pH.n;a B TOqKe:
	a) x = 5;				6) x = 3,5;
	B) x = 4.
					00
1.3.53.	l:I3BecTHO, qTO PM				~ an(z - (1 + i))n B TOqKe z = i CXO.n;HTCH
					n=l
	YCJIOBHO. qTO MO:>KHO CKa3aTb 0 CXO,n;HMOCTH pH.n;a B TOq~e:
	a) z = 1;				6) z = 0;
	B)	l+i
		z = -2-.
1.3.54. CYlllecTByeT JIH CTelleHHoit pH.n;, .n;JIH KOTOPOro BepHo CJIe.n;yIOlllee
	YTBep:>K.n;eHHe:
	a) Ha 060HX KOHllax HHTepBa.JIa CXO,n;HMOCTH pH.n; pacXO,n;HTCH;
	6) Ha O,n;HOM KOHlle HHTepBa.JIa CXO.n;HMOCTH PM CXO,n;HTCH YCJIOBHO,
	a Ha .n;pyroM -			CXO.n;HTCH a6COJIIOTHO;
	B) Ha 060HX KOHllax HHTepBa.JIa CXO.n;HMOCTH pH.n; CXO.n;HTCH a6co-
	JIIOTHO;
	r) Ha O.n;HOM KOHlle HHTepBa.JIa CXO.n;HMOCTH PM CXO.n;HTCH YCJIOBHO,
	a Ha .n;pyroM -			pacXO.n;HTCH;
	~) Ha O.n;HOM KOHlle HHTepBa.JIa CXO.n;HMOCTH pH.n; CXO.n;HTbCH a6co-
	JIIOTHO, a Ha .n;pyroM - pacXO.n;HTCH.

40

	00	(1 + k)	n 2
1.3.55.	HaihH 06JIacTb CXO.u;HMOCTH pH.u;a n~l			(x -l)n.
1.3.56.	CTeneHHblit PM CXO.u;HTCH YCJIOBHO B TO'lKaxZl			= 3 + 2i H Z2 =
	= -1- i. q TO MO)KHO CKa3aTb 0 CXO.u;HMOCTH pH.u;a B .u;pyrHx TO'lKax
	KOMnJIeKcHoit nJIOCKOCTH?

KOHTPOllbHAH PA60TA

BapMaHT 1

1. MCCJIe.u;OBaTb pH.u;bI Ha CXO.u;HMOCTb:						n'
a)	f: n 3 tg2 ~;				00
	n=l	n		6) n~l		3n-t2;
B) n~l (n ~1);				r)	f:	1	.
				n=2 n In n In(In n) ,
.u.)	00	(-l)nn!	.	e)	00	2i + (-l)nn
	E				E	2	•
	n=l 2·5·8· .... (3n - 1)'			n=l		n
2. HaitTH 06JIacTb CXO.u;HMOCTH PMa				00	(2 -	x)n
				E		1
				n=l	n+
3.			00	(z -	2i)n
	HaitTH Kpyr CXO.u;HMOCTH PMa E			~-=3----':­
			n=l		n

BapMaHT 2

1. MCCJIe.u;OBaTb pH.u;bI Ha CXO.u;HMOCTb:										n 3
)	~ arcctg(n + 3) .							00
aL."		2 -	n		3'		6)	E (			1)';
	n=l					n 2		n=l	n + .
	00		~1)					00			1
8) n~l 3~ (n						;	r)	n~l	(n + 1) In(n + 1);
.u.)	00	(_l)n		;			e)	00		'n
	E							E	_z_ .
	n=l n + Fn							n=l n + 2
2. HaihH 06JIacTb CXO.u;HMOCTH PMa							00				(3n + l)n
							E (x + l)n				- 3 - .
							n=l				n
3.						00	(z + i)n
	HaitTH Kpyr CXO.u;HMOCTH PMa E							nH
						n=l	2

41

Bap....aHT 3

1. I1ccJIe.n;oBaTb pa.n;bI Ha CXO.n;HMOCTb:
a)	L: In		-- 2		2) j
	00	(n2		+	-
	n=l	(n	n
	00		+ 1 )n
B) n~l		3n+2			j
~)	00				1
	L:	(_1)n+1 sin -- j
	n=l				nvn
							00	(_1)nxn
2. HaitTH 06JIaCTb CXO.n;HMOCTH pa.n;a							L:		2
							n=l	n+
						00	(z _ i)2n
3. HaitTH Kpyr CXO.n;HMOCTH pa.n;a						L:		1
						n=l	n+
Bap....aHT 4
1. I1ccJIe.n;oBaTb pa.n;bI Ha CXO.n;HMOCTb:								f: 3·5·7· ... · (2n + 1) .
a)	00	arctg2		1			6)
	L:			3 r.:; j				n=l 2·5·8· .... (3n - 1)'
	n=l			V 7!
B	00	(2n + 3)2n.					r)	00		1
								L:		j
)	n~l	n + 2			'			n=l (2n		+ 1)JIn(2n + 1)
	00	(-1)n+1(n+1)						00	n(i + 1)n
~)	L:		2		j		e)	L:	3	.
	n=l		n	+ 3n				n=l		+ z
							00	(n+2)		x3n.
2. HaitTH 06JIacTb CXO.n;HMOCTH pa.n;a n~l								- n -
3. HaitTH Kpyr CXO.n;HMOCTH pa.n;a						00	(z + 2i)n+1
						L:		2
						n=l		n

§ 4. PSlAbl CIlYPbE

PRAbl CIlYPbe

~IIycTb <PYHKIIH.H f{x) - HHTerpHpyeMa.H H nepHo,n;H'IeCKa.HC nepHo,n;OM 271". Ko-

3g'Jg'J'U'4'UexmaM'U tPYp'be <PYHKIIHH f{x) Ha3h1BaIOTC.H '1HCJIaao, aI, a2, •.• , an, ... , bo, bl , b2, ... , bn , ... , KOTOphle HaxO,n;.HTC.H no <PoPMYJIaM

ao = ~ !1T f{x)dx,	(4.1)
1T
an = ~ !f{x)cosnxdx, (n= 1,2, ... ),	(4.2)
-1T

42

"	(4.3)
bn=~ !f(x) sin nx dx, (n=1,2, ... ).
-"
P.aiJO.M iPyp'be <PYHKU;HH f (x) Ha3blBaeTCH PM
00
~o + 2)ancosnx+bnsinnx).
n=l
YcnOBllU1 CXOAMMOCTM pRAa CIlYPbe
PM ct>YPbe HHTel'pHPyeMoil: <PYHKU;HH f(x) Mo:aceT J1H6o	pacXO.n;HTbCH, J1H6o

CXO.n;HTbCH, npH'IeMKaK K <PYHKU;HH f(x), TaK H K <PYHKU;HH, OTJIH'IHOil: OT Hee. YCJIOBHH CXO.n;HMOCTH pH.n;a ct>YPbe 6bl.1IH YCTaHOBJIeHbI HeMeU;KHM MaTeMaTHKOM ,n:HpHX.1Ie.

TeopeMa 1.6 (Alilplilxne).	ECI1Io1 <PYHK~~),! f(x) HenpepblBHa ~Il~ ~MeeT KOHe'lHoe
'l~CIlO TO'leK pa3pblBa Ha	OTpe3Ke [-11", 11"] ~ np~ 3TOM MOHOTOHHa ~Il~ ~MeeT

KOHe'lHOe'l~CIlO 3KCTpeMYMOB Ha [-11",11"]. TO P)'!ACl>YPbe <PYHK~~~ f(x) CXOA~TC),!
All)'!1lI06blX X ~3 [-11", 11"] ~ ero CYMMa paBHa:
1) f(x) All)'!Bcex TO'leKHenpepblBHOCT~ X ~3 ~HTepBalla (-11",11");
2)	l(J(xo - 0) + f(xo + 0»)	All)'! BCex TO'leKpa3pblBa Xo;
3)	l(J( -11" + 0) + f(1I" - 0»)	np~ X = -11" ~ X = 11".

IIycTb f(x) - 'IeTHaH<PYHKU;HH (f(-x) = f(x), \Ix E [-11",11"]). TOl'.n;abn = 0

(n = 1,2, ... ), H, CJIe.n;OBaTeJIbHO, 'IeTHaH<PYHKU;HH pa3JIal'aeTCHB pH.n; ct>YPbe no

KocHHycaM:

f(x) = a 0	00
	+ Lan cosnx,
2
	n=l
l'.n;eao = ~ !" f(x)dx, an = ~!"f(x) cos nx dx, (n = 1,2, ... ).		(4.4)
o	0
AHaJIOl'H'IHOHe'leTHaH<PYHKU;HH f(x)	(T.e. f(-x) = -f(x), \Ix E [-11",11"])	pa3JIa-
raeTCH B PM ct>YPbe no cHHycaM:
	00
f(x) = Lbnsinnx,
	n=l
l'.n;ebn = ~!"f(x) sin nx dx, (n = 1,2, ... ).		(4.5)
o

43

PRA CllYPbe (jJYHK4MM, 3aAaHHOM Ha npOM3BOflbHOM npOMe)l(YTKe

IIycTb f(x) - nepHO,n;H'IeCKaHC nepHo,n;OM 21 cPYHKU;HH, y,n;OBJIeTBopHIOru;aH yCJIOBHHM TeopeMbI )J;HpHXJIe Ha HHTepBaJIe (-I, I). Tor,n;a ee p83JIOlKeHHe B pH,n; clIYPbe HMeeT CJIe.n;yIOru;Hit BH,ll;

00

()

ao

"

(

n7rX

.

n7rx)

= 2+ ~ ancos-l-+bnsm-l-

,

f x

n=l

r,n;e

ao = t /1 f(x}dx,

-I

an = t /1 f(x} cos n;x dx,

bn

= t /1

f(x} sin n;x dx

(n = 1,2, .. .).

-I

-I

PM clIYPbe '1eTHoitcPYHKU;HH f (x) co,n;eplKHT TOJIbKO CB060,n;Hblit 'lJIeHH KOCH-

f(}

00

n7rX

=

ao"

x

2

+ ~ancosT'

n=l

1

1

ao = f/f(x}dx, an = f/f(x} cos n;x dx

(n = 1,2, ... ).

o

0

He'leTHaHcPYHKU;HH f(x} pa3JIaraeTCH B PH,n; cllYPbe no cHHycaM

00

f(x} = L bn sin n;x,

(4.6)

n=l

bn

= f/1f(x} sin n;x dx

(n = 1,2, .. .).

(4.7)

o

1.4.1.Pa3JIO)KUTb B pH.D: fl>YPbe <PYHKIJ;UIO f(x) == 1, 3a,n:aHHYIO Ha UHTep-

BaJIe (-11",11").

Q fl>YHKIJ;UH qeTHaH, n09TOMY OHa pa3JIaraeTCH B pH.D: fl>YPbe no KocuHycaM,

a K09<P<PUIJ;UeHTbI an MO)KHO Haihu no <popMYJIaM (4.4):
7r		7r
2 /	2 /	dx =	217r	2	= 2,
ao = 1i'	f(x) dx = 1i'		1i' 0	= 1i' .11"
o	0	7r
7r
an = fr/ f(x) cos nx dx = fr/			cos nx dx =		2	=
					11"n sin nxl:
o		0
				2		2	(0 - 0)	= O.
			= 11"n (sin 11"n - sin 0)			= 11"n

44

11TaK, ao = 2, an = °(n = 1,2, ...). TaKUM o6pa30M, B ,n:aHHOM CJIY-

qae PH,n: fl>YPbe COCTOUT U3 e,n:UHCTBeHHoro HeHYJIeBOrO CJIaraeMoro, paBHoro

a; = ~ = 1, U pa3JIO)KeHUe UMeeT TPUBUaJIbHblft BU,n:: 1 = 1.

•

Pa3.ll00lCUmb a

pRO

r!Jyp'be

OaHH'bI.e

tjjYH'Il:V,UU,

3aOaHH'bI.e

Ha UHmepaaJl,e

(-11",11"):

1.4.2.

f(x)

= cos2 x.

1.4.3.

f(x)

= sin2 x.

1.4.4.

-1

-11" < x < °

Pa3JIO)KUTb B pH,n: fl>YPbe <PYHKIIUIO f(x) ='{

,

1,

°< x < 11".

a fl>YHKIIUH HeqeTHaH, n03TOMY OHa pa3JIaraeTCH B PH,n: fl>YPbe no cUHycaM.

HaxO,n:UM K03<P<PUIIueHTbI bn no <popMYJIaM (4.5):

~

~

bn = 'frj f(x)sinnxdx = 'frjsinnxdx = -;n cosnxl: =

o

0

2

((_1)n

2

(1- (_1)n)

=

= -1I"2n(cos1l"n - cosO) = -1I"n

-1) = 1I"n

{ a,

n = 2k,

k = 1,2, ...

=

1I"(2k4_1)'

n = 2k -1,

OKOHqaTeJIbHO nOJIyqaeM

f(

x

) = 1. ~ sin(2k -

1)x = 1. (sinx

+

sin3x

+

sin5x

+ ...

)

.

11" LJ

2k -

1

11".

1

3

5

k=l

IIoJIo)KUM B 3TOM paBeHcTBe x = i' Tor,n:a

1=1. (Sin ~ + sin311"2 + sin 511"2

+

) = 1. (1_1 + 1_

)

11"

1

3

5

...

11"

35""

oTKy,n:a i

= 1 - 1+ ~ -

... + (_1)k+1 2k ~ 1 + ..., T. e. MbI nOJIyqUJIU

pa3JIO)KeHUe B 6eCKOHeqHblft PH,n: qUCJIa i' BnepBble 3TO pa3JIO)KeHUe 6bIJIO OTKPbITO 3HaMeHUTbIM HeMeIIKUM MaTeMaTUKOM U <pUJIOCO<pOM JIeft6HuIIeM

(1646-1716).								•
1.4.5.		if\.	A..	f()		=	{3,	-11" < X < 0,
		Pa3JIO)KUTb B pH,n: '¥YPbe'¥YHKIIUIO			x			°< x < 11".
1.4.6.		Pa3JIO)KUTb B PH,n: fl>YPbe <PYHKIIUIO f(x)				= x. -3,
a	fl>YHKIIUH HeqeTHaH, n03TOMY an =			°(n = 0,1,2, ...). OCTaJIOCb onpe,n:e-
JIUTb K03<p<puIIueHTbI bn no <popMYJIe (4.5), T. e.
			~
		bn	= 'frj x sin nx dx.

o

45

,nJUI BblqUCJIeHIUI nOCJIe,II,Hero UHTerpa.rra npUMeHUM MeTO,n: UHTerpUpOBa-

HU5I no qacT5IM. TIO.TIOXUM 'U	= x, dv = sin nx dx. Tor,n:a du = dx, v =
= / sin nx dx = - kcos nx, oTKy,n:a
/ ~ xsinnxdx = -kxcosnxl: + k/~cos nx dx =
o		0
= _l(7rcos7rn -	0) + lSinnxl~ = _7I(_1)n = 7I(_1)n+l.
n		n2	0	n	n
OKOHqaTe.TIbHO nO.TIyqaeM bn = *.		~(_1)n+1 = ~(_1)n+1, CTa.rrO 6bITb,
f(x) = X = 2f:(-1)n+~ sinnx = 2		(si~x _	Si~2x + sin33x _		sin44x + .. .).
n=l

TIo,n:CTaBUB 3HaqeHUe x = ~ B 9TO paBeHcTBo, npu,n:eM K yxe BCTpeqaBmeMYC5I HaM B 3a,n,aqe 1.4.4 p5l,!l;y JIeft6Huu;a

sin 7!:

•

sin 371".

+

)

".TI"

7I = 2 ( __2

sm 7r + __2__

sm 27r

...

2

1

2

3

4

,I<

I<

i

\

= 1 -l + ~ - ... + (-1 )k+1 2k ~ 1 +...

•

Pa3.1/,OO/C'Um'b a PSiO fPyp'be tjjYH'K'U,'UU,

3aoaHHwe Ha UHmepaa.l/,e (-7r, 7r):

1.4.7.

f(x) = 1- 2x.

1.4.8.

f(x) = Ix -

3.

2

-7r < X

< 0,

1.4.9.

Pa3JIOXUTb B P5l,n: <l>YPbe <PYHKU;UIO f (x) = {

2

'

0

< x < 7r.

-4,

Q <l>YHKU;U5I 06IIIero Bu,n:a,

n09ToMY K09<p<puu;ueHThl

<l>YPbe

HaxO,n:uM

ne

<popMY.TIaM (4.1)-(4.3):

~

0

~

*x[~ -

ao = } / f(x) dx = } / 2dx + }/(-4) dx =

#xl: = 2 -

4 = -2,

-~

-~

0

~

0

~

an =

} / f(x) cosnxdx =

} / 2 cos nxdx +}/(-4) cosnxdx =

-~

-~

0

1 sill. nx I~ - 0

_

0

~sin nx 1 _

,

-

7r

n

_~

7r

n

0

-

~

0

~

bn = } / f(x) sinnxdx = } / 2sinnxdx + } /(-4) sinnxdx =

-~

-~

0

= - ;n cosnx[~ + 7r~cosnxl: = - ;n (1 -

(_1)n) + 7r~(( _1)n -

1) =

46

					n = 2k,	(k = 1,2, ...).
					n = 2k -1,
B HTore HMeeM
		ao		~.	12 ~ sin(2k -	1)x	•
	f (x) = 2		+ L.J bn sm nx = -1 -		7r L.J 2k _ 1 .
				n=l	k=l
1.4.10.	f(x)	= {	5,	-11" < X < 0,
				0< x < 11".
		-3,
1.4.11.	f(x) = { -7,			-11" < X < 0,
				o<x < 11".
			2,
1.4.12.	f(x)	= x2 •		1.4.13. f(x) = Ixl.

1.4.14.MCnOJIh3YH pa3JIO:lKeHHe H3 3a,n,aqH 1.4.12, BhIQHCJIHTh CyMMy pH,n:a

00 1

L2"·

n=l n

1.4.15.lIPH nOMOIII.H pa3JIO:lKeHHH H3 3a,n,aQH 1.4.13, HaiiTH CyMMy pH,n:a

00 1

k;;'l (2k - 1)2

1.4.16.f(x) = 1 - ~Ixl.

1.4.17.	f(x) = sin ax (a - He I.J;eJIoe QHCJIO).
1.4.18.	f(x) = {-x,			-11" < X ~ 0,
		0,		0 < x < 11".
1.4.19.	C nOMoru;hIO			pa3JIO:lKeHHH H3 3a,n,aQH 1.4.18, HaiiTH CyMMy pH,n:a
	1	1		1	+ ....
	1 + 32	+ 52	+ ... + (2k _ 1)2

1.4.20.Pa3JIO:lKHTh B PH,n: q,YPhe no cHHycaM Ha OTpe3Ke [0,11") <PYHKI.J;HIO

X,	0 ~ x <~,
f(x) = {	11"
11" -x,	'2~ x ~ 11".

Q lIPO,n:OJ1:lKHM <PYHKI.J;HIO Ha oTpe30K [-11",0) HeQeTHhIM 06Pa30M (pHC. 1).

Tor,n:a nOJIYQeHH8JI <PYHKI.J;ua HeQeTHaH H ee pH,n: q,YPhe co,n:ep:lKHT TOJIhKO

47

y

x

Pu.c. 1

CHHYCbI. Hat!:,!l;eM K09cPcPHIIHeHTbI bn

(n = 1,2, ...):

7r

7r

7r

f(x) sin nx dx =

"2

x) sinnxdx =

b = *f

*f

xsinnxdx + *f(1r -

n

o

0

1!:

= *( -X C~nx I:7r

+ *.1r f

2

*f7r x sin nx dx =

+ kf~COS nx dx)

7r sin nx dx -

o

1!:

1!:

2

2

= ~(_ 7rCOS ~n + sinnx I~) + 2(- cosnx 17r)-

1r

2n

n 2

0

n

~

7r

7r

)

cos 1I. n

+

2 sin 1I. n

_ ~ ( _xcosnx 1 + Ifcosnxdx

= ___2_

2

1r

n

1!:

n

7r

n

1rn2

2

7r

"2

7r

_ 2cos1rn

-.L.

17r-

2 cos"2 n

2cos1rn _ cos"2n _

n

+

n

+

n

n

1rn

2 smnx 7r -

{O'

"2

=

2sin~n

+

2sin~n

=

4sin~n

=

n=2k,

1rn2

1rn2

1rn2

4( _l)k+l

,

n = 2k -

l.

1r(2k - 1)2

4

00

(_l)k+l sin(2k -

l)x

T8J{HM 06pa30M, f(x) = 1f

E

(

)2

.

k=l

2k -

1

1r

1r

4 (1

1

1

+...+

1

+...

)

,oTKY,!l;aeIIIe

npH X = "2 HMeeM

"2 =

1f

+32

+ 52

(2k _ 1)2

1

1

+...+

1

7r

2

•

pa3 HaxO,!l;HM, 'ITOcyMMa p.H,!l;a 1+2" +2"

2 +... paBHa -8 .

3

5

(2k -

1)

1.4.21.Pa3JIO)KHTb B p.H,!l; fl>YPbe no cHHycaM Ha HHTepBaJIe (0,1r) CJIe.Ll:Y-

IOIIIHe cPYHKIIHH	6) f(x) = ~ - ~.
a) f(x) = x;

1.4.22.Pa3JIO)KHTb B p.H,!l; fl>YPbe no KocHHycaM Ha OTpe3Ke [0, 1r) cPYHKIIHIO

f(x) = ~ - ~'.

48

y

71'

4

Puc. 2

a IIpo,n:oJDKUM

,n:aHHYIO <PYHKD;UIO Ha oTpe30K [-11',0] qeTHbIM o6pa30M

(puc. 2). B

pe3YJIbTaTe nOJIyqUTC.H qeTHM <PYHKD;U.H, P.H,n: <IIYPbe KOTOpoii co-

CTOUT TOJIbKO U3 KOCUHYCOB. BbIqUCJIUM K09<p<puD;ueHTbI an (n = 0,1,2, ...)

no <popMYJIaM (4.4):

71"

71"

ao = *Jf(x)dx ~ *JUi -~) dx =

o

0

= *(~x_x;) I: = *(71';

-

71';)

= 0,

71"

71"

an = *Jf(x)cosnxdx = *J(~ -~) cos nx dx =

o

0

71"

71"

71"

= !Jcosnxdx -

IJxcosnxdx = sin nx 171" _

x sin nx 171"

+ ...l..Jsin nxdx =

2

71'

2n

0

71'n

0

71'n

000

= -

1 ( .

. 0)

-

-

1 ( .

0'

0)

-

cos nx 171"

=

sm7l'n-sm

71'sm7l'n-

·sm

--

0

2n

11'n

{O,

11'n2

1

1

n = 2k,

= --(cos11'n - cosO) = - (1 - (_I)n) =

2

n = 2k - 1.

11'n2

71'n

2

- ,

71'n

2

IhaK, f(x)

2 00 cos(2k - l)x

• IIOJIO:lKUM B 9TOii <popMYJIe x = 0. Tor,n:a

= IT

L:

1)

2

k=l (2k -

71'

2

00

(

1

00

(

1

1

1

71'2

-4

= IT

L:

1)

2'oTKy,n:a

L:

1)

2 = 1 + 2'+

2'+ ... = -8 ' 'ITO

k=l

2k -

k=l

2k -

3

5

COBna,n,aeT C Haii,n:eHHbIM paHee 3HaqeHUeM .D:JI.H CYMMbI 9Toro p.H,n:a.

•

1.4.23.Pa3JIO:lKUTb B P.H,n: <IIYPbe no KocuHycaM Ha oTpe3Ke [0,11'] CJIe,n:yIO- r:u;ue <PYHKD;UU

a)f(x)

B)f(x)

=

=

-x;	6)	f(x) = ~x - 1;
_x2 ;	r)	f(x) = ~X2 + 3.

1.4.24.Pa3JIO:lKUTb B P.H,n: <IIYPbe <PYHKD;UIO f(x) = x Ha UHTepBaJIe (-3,3).

49