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Владимирский государственный университет им. Столетовых

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30	fUNKCIONALXNYE RQDY

PRI n 2 N IMEEM

; x! sin nx dx = ;

; x! d cos nx =

bn =

; x! cos nx

; x! =

= ;

cos nx d

= cos n + 1 +

sin nx

= 1 + (;1)

= 8 k

n = 2k k 2

n n2

0 n = 2k

;

1 k

sin 2kx

:: .

f(x) S(x) = 1

k=1

2.3.5. rAZLOVITX FUNKCI@ y = 8

PRI 0 x <

W RQD fURXE

NA (; ) PRI ` = .

PRI ; < x < 0:

/ wY^ISLQEM a0 =

f(x) dx =

Z (;x) dx +

2x dx = 2

;

an = 1

Z f(x) cos nx dx =

(;x) cos nx dx + 1 Z

2x cos nx dx =

;

= "INTEGRIRUEM PO ^ASTQM PRI u = x du = dx dv = cos nx dxv = n1 sin nx# =

! +

! =

= ;

n sin nx

cos nx

n sin nx

cos nx

;

= ;

(1 ; cos n) +

(cos n ; 1) = ;

(1 ; (;1) )

bn =

f(x) sin nx dx = : : : = (;1)n+1 n

;

n+1 1

f(x) S(x) =

4 +

; n2

(1 ; (;1) ) cos nx +

(;1)

n sin nx: .

n=1

2.3.6. rAZLOVITX FUNKCI@ y = x W RQD fURXE NA (;1 1) PRI ` = 1.

zADA^I DLQ SAMOSTOQTELXNOGO RE[ENIQ

/ tAK KAK f(x) = x { NE^ETNAQ FUNKCIQ, TO an = 0 DLQ WSEH n I

bn = 2 Z x sin nx dx = ;

x d cos nx =

= ;

x cos nx

cos nx dx =

cos n

(

1)n+1

= ;2

sin nx

= 2

; n

2n2

f(x)

S(x) = 1 (;1)n+12 sin nx: .

n=1

3.zADA^I DLQ SAMOSTOQTELXNOGO RE[ENIQ

iSSLEDOWATX NA SHODIMOSTX RQDY.
	1											n			1												1				1																	1				1														1							1
3.1.				(;1)										;			.				3.2.					n=1					10n .						3.3											n=1				2n.								3.4.					n=1					3n + 2.
	n=1																									n=1																						n=1																	n=1
3.5.	1X					1				.			3.6.								1					nX						.		3.7.								1						Xn										.		3.8.		1 X							(n!)2														.

			e			1=n																			2																						10 + 3n
	n=2		e																		n=1				n + 100															n=1							10 + 3n															n=1					(2n + 1)!
	X																				X																				X																					X
	1 2nn!																				1				4nn!												1						pn!																	1			1=3																2
3.9.	nX=1			nn							. 3.10.										nX=1 nn							.		3.11.								nX=1					3n)						. 3.12.											nX=1 n;							sin					n;									.
3.13				1									p			n					.				3.14.									1											1						.									3.15.									1 ne;n.
3.13										(n + 1)2											.				3.14.															n						sin n					.									3.15.									1 ne;n.
			n=1							(n + 1)2																										n=1				n						sin n																					n=1
				X																						ln n										X						;					2n								1													X
	1				pn + 1																				1	ln n																;					2n								1															n5 + 1
3.16.	1															. 3.17.									1				5 .					3.18.							1									;							2	.		3.19.		1 ln																						.
3.16.							3				+ 5					. 3.17.											n		5 .					3.18.																;							2	.		3.19.		1 ln											n			5								.
	n=1					n					+ 5														n=1		n													n=1						(n + 1)																n=1											n
	X			1							arctg n														X									1							X																					X											1						1
				1							arctg n																							1						1=n																																	1						1
3.20.			nX=1 10n												; n					.					3.21.									nX=1			3							; 1 sin n.																3.22.											nX=1							n2!					.
		1						3n + 1																												n + 1																									7												1
								3n + 1																												n + 1																									7												1
3.23.																		.			3.24.								1 ln															.				3.25.												1 pn					ln cos n													! .
3.23.	n=1										n!							.			3.24.								1 ln							2n + 5								.				3.25.												1 pn					ln cos n													! .
	n=1																											n=1																																n=2
		X																										X																																X
	1													1																1														1 10																					1													3
																																																																														3
																																																								pn															3						pn
3.26.						p													p					arcsin n.										3.27.								n=1 n e;																	.	3.28.					n=1 n e;																			.
									n + 1
	n=1								n + 1													n
	X																;								1		ln n																X					1 n + 1																	X			1												1
3.29.		1 ln n											.				;								1		ln n						.	3.31.														1 n + 1											.	3.32.								1					sin							1				.
3.29.								en					.					3.30.										n!					.	3.31.																				n!					.	3.32.													sin						n					.
	n=2							en																	n=2			n!																		n=1								n!													n=1												n
	X																								X									1													X																				X								1
3.33.					1 arctg2 n															.					3.34.									1							n + 1													.						3.35.							1								1							.
3.33.			n=1										n3							.					3.34.									n=1				n3			+ n2 + 1													.						3.35.						n=2						n ln n										.
			X																	1			n2											X					sin2 (np														)														X						1
			X																	1			n2											X					sin2 (np											n			)														X						1
3.36.		1				4;n								1 + n!											.	3.37.								1										npn											. 3.38.										1																		.
																																																																					p					ln n
																																			n=1																													n=1					p			n		ln n
	n=1																																		n=1																													n=1
	X																																			X																												X

zADA^I DLQ SAMOSTOQTELXNOGO RE[ENIQ

)

1 ln(n)

+ 2

3.39.

arctg(n

3.40.

3.41.

n(n + 1)(n + 2)

n2(2 + sin n)

n=1

n2 + 5

3.42.

n + 4 tg pn. 3.44.

1 pn 1 ; cos n + 1!. 3.43.

1 ln n2 + 4.

n=1

3.45.

3.46.

n=1 pn + 4 sin n + 1.

n=1 arcsin(n2 + 3)5=2 .

n + 1

2n+1 n3 + 1

(n + 1)!

3.47.

n=1

2n(n

;

1)!.

3.48.

n=1

(n!)2 .

3.49.

n=1

103n!

: : :

(3n

3.50. 1

(2n)! . 3.51. 1

2n+1n!

;

. 3.52. 1 3;n

n + 1

n=1

2n2 + 1

3.53.

1 n4

. 3.54.

0 n2 + 1

. 3.55.

3n + 5

(n2

;

2) ln(2n)

n=1

n=2

3.56.

n=1 n

arctg

3.57.

n=5

n ln2(3n

;

11)

. 3.58.

n=4

n +;

! .

2n + 1

1=n

(3n

3.59.

. 3.60.

;

n + 3

(5n 3)

;

n=1

iSSLEDOWATX NA ABSOL@TNU@ I USLOWNU@ SHODIMOSTX RQDY.

3.61.

1 (;1)pn+1 .

3.62.

1 (;1)n .

3.63.

1 (;1)n+1p

ln n

n + 2

n=1

n=2

n=1

3.64.

(

1)n

n ;

3.65.

1 (

1)n+1 tg

;

pn(n + 1)

;

npn

n=2

n=1

3.66.

(

1)n

3n + 1

3.67.

(;1)n .

3.68.

(

1)n+1n10e;n.

; 4n + 5!

n1=n

;

n=0

n=2

n=1

(;1)n+1

3.69.

(

1)n+1

3.70.

. 3.71.

(

1)n ln n.

;

3n + 2

100n + 1

n=1

n=2

3.72.

(

1)n

. 3.73.

(;1)n+1

. 3.74.

1 (

1)n+1

n ; 1

n=2 ;

n=1 2n

;

arctg n

n=1 ;

n2 + 1

n ln n

arctg n

n+1

3.75.

(

1)n+1

3.76.

(;1)

ln n. 3.77.

(;1)

;

npn

n=1

n=2

n=1

n + 1

3.78.

(

1)n+1

2n + 1

. 3.79.

(;1)n+1

. 3.80.

(;1)n

;

n(n + 1)

n X

n=1

n=2 ln(n + 1)

n=2 n ln n (ln ln n)

3.81.

1 (;1)n+1

! .

3.82.

1 (;1)n

+ 1

(n + 1)!

n=1

3.83.

(;1)2

;

(3n

5) ln (4n

15)

n=5

zADA^I DLQ SAMOSTOQTELXNOGO RE[ENIQ

nAJTI OBLASTI SHODIMOSTI FUNKCIONALXNYH RQDOW.																																																																												1 (1)n
3.84.	1								n	.			3.85.													1			5	n		n					3.86.													1									n		,	a				=				0.					3.87.												.
3.84.						x				.			3.85.																5		x .						3.86.																(ax)								,	a				=				0.					3.87.												.		xn
	n=0																									n=0																						n=0																		6																n=0							xn
	X				1																					X														1									X																									1 0								X						1			n
													1																																					n																													x(x + n)												n
3.88.																								.			3.89.																														.					3.90.																													.
3.88.										x			2n			+ 1								.			3.89.																	3			nx				;					1	.					3.90.																				n									.
			n=1							x						+ 1																							n=1					3												1																n=1										n
			n=1																																				n=1																																	n=1
	1			Xn							+ 1																		1								n			X																		n										1						X																	n
3.91.	1					x					+ 1						.				3.92.								1								n								1 + 2x														.			3.93.						1							1@ 1 + 2xA																	.

									xn																								n2				+ 1 1 + 3x!																																						5


n=1																													n=1																																						n=1								npn 4 + x !
	X									1 (																			X																								n.														X												1 (x ; 1)n .
3.94.																			1)n								n		; 1														x																			3.95.
																										5
																										5
									n=2						; pn(n + 1) 3x 1!																																																														n=1					(x + 1)n
									X													n																									n																															X
	1								(x								1)																				1							;																						1				1															1			n
									(x				;				1)																											x																										1																		n
3.96.																							n		.		3.97.																										.				3.98.																.		3.99.																	.
3.96.																							n		.		3.97.																2n				+ 1						.				3.98.												x		n		.		3.99.													x		n		.
	n=1						n(x + 1)																												n=1							x					+ 1																	n=1					x														n=1					x
	X										1																					1				X				1																						1			X																			X
				1							1																					1								1																						1				1																	1			ln n
3.100.																.						3.101.																						.						3.102.																x .						3.103.																	x .
3.100.														n		.						3.101.																	2				n	.						3.102.																x .						3.103.															n		x .
			n=1							nx																					n=1							n x																								n=1			n																	n=1					n
				X										xn																		X																			n3x2n											X																					X				1)n
							1							xn																												1									n3x2n																												1			(x					1)n
3.104.							1																	.			3.105.															1																				.			3.106.														1					;n .
3.104.														n								1		.			3.105.																			(n + 1)												3	3	n		.			3.106.																			;n .
					n=1							x										1																		n=1						(n + 1)													3																		n=1							n2
					n=1											; n																								n=1																																					n=1
	1					X										; n										3.108. 1																X													n											1				7		n			(x			X n
3.107.	1								(x				;				1)								.														n!(x + 5)																	. 3.109.										1				7					(x	; 7) .
													;																																9n
								ln(2 + n)
	n=1							ln(2 + n)																									n=1																																	n=1									n!
	X																																	X																																X
rAZLOVITX FUNKCI@ W RQD PO STEPENQM x, T.E. W RQD tEJLORA W OKRE-
STNOSTI TO^KI x0 = 0.
											1																																																																																2
3.110.											1							.								3.111.												sh x.													3.112.												ch x.												3.113.										xe2x .
3.110.							5 + 7x											.								3.111.												sh x.													3.112.												ch x.												3.113.										xe2x .
3.114.					p				x											.						3.115.											sin2 x.														3.116.											cos2 x.													3.117.										sin3 x.
3.114.					p				9 + x2											.						3.115.											sin2 x.														3.116.											cos2 x.													3.117.										sin3 x.
3.118.							1							. 3.119.																1					. 3.120.																					5x			; 1							. 3.121.															2x + 3											.
3.118.											2			. 3.119.																		4			. 3.120.																			2					; 1				6			. 3.121.														2												.
		1 ; x																										1 + x																								x				; 5x +							6															x			;		4x + 3
		1 ; x																										1 + x																								x				; 5x +																						x			;		4x + 3
						x							7																																																	5 1 + x																					1 + 3x
3.122.	6								;						x2						. 3.123. ln(1																;				x2).							3.124. ln v																			x			. 3.125. ln													1					3x				.
	6			;					x		;				x2																						;																									u1			;		x																1		;			3x
				;							;																																																			t			;																				;
3.126. ln(1 + 5x). 3.127. ln(5 + 2x). 3.128. ln(1 + 2x2). 3.129. ln																																																													1 + 2x.
																																																	x sin x																																		1 + 3x
					ln(x2 ; 5x + 4).																																												x sin x																														x arctg x
3.130.					ln(x2 ; 5x + 4).																											3.131.															Z								x				dx.						3.132.												Z						x					dx.
		x					x																									x																0																												0
		x					x		; 1 dx. 3.134.																							x		ln(1 + x) dx.
3.133.	Z				e				; 1 dx. 3.134.																						Z			ln(1 + x) dx.
	Z								x																						Z										x
	0																														0
rAZLOVITX FUNKCI@ W RQD PO STEPENQM x ; x0,																																																															T.E. W RQD tEJLORA W
OKRESTNOSTI TO^KI x0.																																																														1

			3																																				x																																																p
			3																																				x
3.135. x						; x, x0 = ;1. 3.136. e , x0 = 2. 3.137. x, x0 = ;3. 3.138.																																																													x,
3.135. x						; x, x0 = ;1. 3.136. e , x0 = 2. 3.137. x, x0 = ;3. 3.138.																																																													x,

zADA^I DLQ SAMOSTOQTELXNOGO RE[ENIQ

x0 = 4. 3.139. sin

, x0

= 2. 3.140.

, x0

= 1. 3.141. sin x, x0 = 2 .

2x + 3

3.142. cos x, x0 =

2 .

w ZADA^AH 3.143{3.151 RAZLOVITX W RQD fURXE FUNKCI@ f(x) PRI ` = .

3.143.

f(x)

;

3.144.

f(x) = 8

; < x < 0

< x

x < :

3.145. f(x) =

x2. 3.146. f(x)

= sin3 x.

3.147.: f(x)

cos3 x.

3.148. f(x) =

sin4 x.

3.149. f(x) =

cos4 x.

3.150. f(x)

= jxj.

3.151. f(x) = 8

;

x < 0

9 3.152. iSPOLXZUQ 3.150, WY^ISLITX

SUMMU RQDA

(2n

;

1)2

n=1

(;1)

n+1

3.153. iSPOLXZUQ 3.151, WY^ISLITX SUMMU RQDA

n=1

3.154. iSPOLXZUQ 3.145,

WY^ISLITX SUMMY RQDOWX1

;I 1 (;1)2n+1 .

n=1

w ZADA^AH 3.155{3.157 RAZLOVITX W RQD fURXE FUNKCI@ f(x) DLQ

UKAZANNYH `.

3.155. ` = 1, f(x) =

x . 3.156. ` = 2, f(x) = 8

0 < x

3.157. ` =

2 < x <

2, f(x) = 8

< x

;2 2 < x < 4:

3.158.

pRI

= 1 RAZLOVITX FUNKCI@ f(x) = 1

;

x NA INTERWALE

(0 1): A) PO KOSINUSAM KRATNYH DUG (\TO OZNA^AET, ^TO MY DOOPREDELQEM

FUNKCI@ f(x)

NA INTERWALE (;1 0) TAK,

^TO f(x) STANOWITSQ ^ETNOJ

FUNKCIEJ NA

INTERWALE

(;1 1)),

B) PO

SINUSAM

KRATNYH DUG (\TO

OZNA^AET, ^TO MY DOOPREDELQEM FUNKCI@ f(x) NA INTERWALE (;1 0) TAK, ^TO f(x) STANOWITSQ NE^ETNOJ FUNKCIEJ NA INTERWALE (;1 1)).

oTWETY. w ZADA^AH 3.1, 3.3{3.7, 3.10, 3.11, 3.14, 3.18, 3.24, 3.26, 3.32, 3.35, 3.38, 3.41, 3.51, 3.54, 3.55, 3.58, 3.59 RQDY RASHODQTSQ. w ZADA^AH

3.2, 3.8, 3.9, 3.12, 3.13, 3.15{3.17, 3.19{3.23, 3.25, 3.27{3.31, 3.33, 3.34, 3.36, 3.37, 3.39, 3.40, 3.42{3.50, 3.52, 3.53, 3.56, 3.57, 3.60 RQDY SHODQTSQ.

w ZADA^E 3.61 RQD SHODITSQ ABSOL@TNO PRI p > 1, SHODITSQ USLOWNO

PRI 0 < p 1, RASHODITSQ PRI p 0. w ZADA^AH 3.62{3.64, 3.70, 3.71, 3.73, 3.75, 3.77{3.80 RQDY SHODQTSQ USLOWNO. w ZADA^AH 3.65, 3.66, 3.68, 3.72, 3.76, 3.81{3.83 RQDY SHODQTSQ ABSOL@TNO. w ZADA^AH 3.67, 3.69, 3.74

zADA^I DLQ SAMOSTOQTELXNOGO RE[ENIQ

RQDY RASHODQTSQ.

3.84:

jxj

3.85:

jxj

< 1=5.

3.86:

jxj

1=jaj. 3.87:

jxj

3.88:

jxj

3.89:

3.90:

jxj

3.91:

2 ;5

. 3.93: x

2 "4

. 3.95: x

3.92: x =

. 3.94: x =

3.96: x

3.97: x

3.98:

3.99:

3.102:

3.100: (;1 ;1] S(1 +1).

3.101:

jxj

3.103: x > 1. 3.104:

jxj

< 1. 3.105:

jxj

3.106:

x < 3.

3.107:

;

3.108:

;

3.109:

(

(;1)n

x2n+1

3.110:

5n+1xn ,

jxj

< 5=7. 3.111:

(2n + 1)!,

n=0

x2n

n=0

2n 2n+1

< +1. 3.112:

(2n)!,

;1 <

+1.

3.113:

n! x

n=0

+ .

3.114:

x +

(

1)n (2n ; 1)!!x2n+1,

; n!2n32n+1

j j

n=1

n+1 22nx2n

n 22n;1x2n

3.115.

(;1)

(2n)! , jxj < +1. 3.116. 1 +

(;1)

(2n)!

, jxj <

n=1

32n ; 1

1 x2n, x <

. 3.117. 3

(

1)n+1

x2n+1,

x < + .

3.118.

;

(2n + 1)!

j j

4 n=1

n=0

3.119.

n=1(;1)

jxj < 1.

3.120.

n=0

2n+1

;

3n+1

! x

jxj

< 2.

1 1X

, jxj

3.121.

2 nX=0 5 ;

;

< 1.

3.122. n1=0

"(;1)n+1

;

# xn, jxj < 2.

3n+1

2n+1

2 1 x2n+1

1 x2n

3.123.

;

n ,

jxj

3.124.

jxj

n=0

2n + 1

1X32n+1

2n+1

X n+1 5n n

3.125. 2

2n + 1 x

, jxj < 3. 3.126.

(;1)

n x , ;5 < x 5.

n=1

n=0

n+1

3.127. ln 5 +

(;1)

n5n x , ;2 < x 2.

n=1

n+1 2n 2n

n 3n

;

2n n

3.128.

(

;

j p2

. 3.129.

(

;

x ,

< x

n=1

;1 x < 1.

3.130. ln 4 ; nX=1 1 + 4;

n ,

x2n+1

3.131.

(;1)

(2n + 1)(2n + 1)!, jxj < +1. 3.132.

(;1)

(2n + 1)2 , jxj 1.

n=0

36 rQDY

n+1 xn

3.133.

n=1

n!,

jxj < +1.

3.134.

n=1

(;1)

n2 ,

jxj 1.

3.135. 2(x + 1) ; 3(x + 1) + (x + 1)

. 3.136. e;

1 (x + 2)

, jxj < +1.

n=0

1 (x + 3)n

3.137. ;

, ;6 < x < 0.

n=0

3n+1

1 (

3.138. 2 + x ; 4

;

1)n+1 1 3

(2n

;

3)(x

;

4)n, 0

n=2

23n

1n!

;

3.139:

(

< x < + .

(2n)!

n=0

;

3.140: 1 (;1)n

(x ; 1)n, ;

< x <

5n+1

n=0

(;1)n

2n,

3.141:

;

< x < +

n=0

(2n)!

(;1)n+1

2n+1,

3.142:

< x < +

;

2 !

n=0

(2n + 1)!

1 (

1)n sin nx. 3.144:

cos(2n ; 1)x

3.143: + 2

sin nx

nX=1 ;

nX=1 2;

(2n

;

1)2 + (

;

1)n+1

3.145:

+ 4

(;1)

n cos nx

n2 .

3.146:

4 sin x ;

4 sin 3x.

n=1

;

3.147:

4 cos x + 4 cos 3x.

3.148:

2 cos 2x +

8 cos 4x.

1 cos(2n

1)x

3.149:

2 cos 2x +

8 cos 4x.

3.150:

;

n=1

(2n

;1)2

;

1 sin(2n

;

3.151:

1)x. 3.152: 2=8. 3.153: =4. 3.154: 2=6, 2=12.

n=1

;

1) x

(2n

1) x

cos(2n

3.155:

;

;1)2

. 3.156:

2 +

sin

n=1

(2n

n=1

;

(2n

1) x

;

3.157: ;1 +

sin

n=1

;

1) x

1 sin n x

cos(2n

3.158: A)

;1)2

n=1

(2n

n=1

;

4.kONTROLXNYE ZADANIQ

w ZADA^AH (1){(6) ISSLEDOWATX ^ISLOWYE RQDY NA SHODIMOSTX. w ZADA^AH (7){(9) NAJTI OBLASTI SHODIMOSTI RQDOW.

kONTROLXNYE ZADANIQ

w ZADA^AH (10){(11) RAZLOVITX f(x) W RQD PO STEPENQM x I NAJTI f(10)(0)

I f(11)(0):

4.1.

(1)

n3 + 2

(2)

1 arcsin

n=1 n2(2 + sin n)

(n2 + 3)5=2

n=1

1 1 4 7

1 n4arctg2n

(3)

: : :

(3n ; 2) (4)

(5)

(n2

2) ln(2n)

n=1

2n+1n!

n=1

n=2

;

(n + 1)

(6)

n=1(;1)

(n + 1)!.

(7)

(n2

+ p

+ 1)x+1

(8)

2n + 1

n=1

(9)

. (10)

(11) Z

cos

100x

dx.

n=1 n3

(x2

+ 7)n

;

sin2 (npn)

n + 1

(2) 1 pn 1 ; cos

(3) 1

4.2.

(1)

npn

n=1

n + 1

2n(n

;

1)!

n=1

n+1

2n + 1

(4)

n=1

n + 1

(5)

n=5 n ln2(3n

;

11)

(6)

n=1(;1)

n(n + 1).

(7)

(8)

1 (n ;

(x + 3)2n

(9)

xn + 1

(x2

6x + 13)

n=1

2n + 3

n=1

;

(10)

(11)

Z e;6x dx.

;

1 arctg2 n

4.3.

(1)

(2)

tgp

(3)

n + 4

(n!)2

n=1

n+1

! .

(4)

n=1 n

3n + 5

(5)

n=5 n ln2(2n

;

(6)

n=1(;1)

2n + 1

n+1

(7) 1 (;1)

(8)

(;1)

3)n (9)

sin3n x. (10)

(n + 1)5n

;

n=1 n2

n=1 nln(1+x)

n=1

;

(11) Z sin

100x2

dx.

arctg(n2)

n2 + 5

2n+1

n3 + 1

4.4.

(1)

(2)

1 ln n2

(3)

(n + 1)!

n=1

n(n + 1)(n + 2)

+ 4

n=1

(4)

2n2

+ 1

(5)

(6)

(;1)n+1

n=1 0 n2 + 1

n=5 (2n + 3) ln2(2n

;

n=2 ln(n + 1)

X n

(7)

(8)

(x ; 1)

(9)

1 +

3n=(x;1).

x+2

n9n

n=1

(pn + pn + 1)

n=1

(10) ln

; x

; 6x2

(11) Zx cos x2 dx.

rQDY

4.5.

(1)

1 ln(n)

(2)

sin

(3)

103 n!

n=1

n + 4

n + 1

n=1 (2n)!

(

1)n

(4)

n=1

1 + n!

(5)

n=5 (3n

5) ln2(4n

15)

(6) n=3

n ln n;(ln ln n)

;

n en(x2;4)+xp

(;1)n(n + 1)(x + 7)n

(7)

1 +

(8)

n=1

(n + 3)22n

;

1 n + 1

2 x

(9)

;

4x + 6

. (10) 2x cos

;

x (11)

dx.

n=1 3n

Z p1 + x4

4.6.

(1)

1 5

; sin n

(2)

arctg

(3)

(2n + 2)!

n=1 n

n=2 n

n=1 2n(3n + 5)

ln(n)

;

2n2

2n + 1

(4)

n=1

(5)

n=2 (3n + 4) ln2(5n + 2)

(6)

n=1(;1)

n2 + 1

n 1

;

1 (

;

(7)

n=1

+ n2

(8)

n=1

;2n

(x ; 2)

(9)

n=1

n sin

(10)

e2x

; e;2x

;

2 (11)

; e;2x dx.

1 ln[(n2 + 3)=(n2 ; n)]

4.7.

(1)

1 (1 ; cos(n))=(n3 + 2)

(2)

n=1

n=2

1 n + 5

2n + 2

(3)

n=1

sin

(4)

n=1

3n + 1

(5)

n=1 (2n + 1) ln (n

5 + 2)

(;1)n

5)2n+1

(6)

(7)

1 en2 sin[(x2+1)=n]

(8)

1 (x

;

n=2 (n

+ 1) ln n

n=1

3n + 8

(9)

2n=(4;x). (10)

(11)

x ln(1 + x=5) dx.

n(n + 1)

12 + x

n=1

1 n(2 + cos(n ))

1 1

1)=n3

4.8.

(1)

n=1

2n2

;

(2)

n=1 e

;

; 1

(3)

n=1

arctg n

(;1)n

(4)

;

(5)

(6)

n=1

5n + 1

n=1

n ln(n + 1)

n=1 (n

2 + 1) ln (n

3 + 1)

X n + 5

(

1)n

(7)

(8)

;

(x + 6)n (9)

n3x

n(x2

5x + 10)

n=1

(n + 3) ln(n + 3)

n=1

;

(10)

(11)

dx.

p27

;

p27 + x3

1 p

4.9.

(1)

(3 + sin n)=

; n

(2)

n arcsin [(n + 1)=(n

; 2)]

n=1

n=2

kONTROLXNYE ZADANIQ

(3)

1 nn=(3nn!)

(4)

1 (n=(10n + 5))n2

(5)

1 (n ; 2);1 ln;1(n ; 3)

n=1

n=5

(;4

1)n+1

1 n arcsin 3;nx

;

6)n

(6)

(7)

(8)

n=1 np2n + 3

n=1

n=1 (n + 2)3n

(11) Z

3x3

(9)

sin

(2x). (10) ln

1 + x ; 6x

dx.

n=1

sin2 n

4.10.

(1)

n=1 n2 + 1

(2)

n=1 n e;

; 1 sin

n + 1

(3)

n=1

(2n)!

(4)

1 n arcsinn

(5)

(6)

p(;1)n

sin

n=1 (2n

1) ln(2n)

2pn

n=1

3n + 1

1 (x + 5)2n;1;

1 lnn(x + e)

(7)

(8)

1)4n (9)

. (10) (x ; 1) sin 5x

n=1

1 + x2n

n=1

(2n

n=1

+ e

;

(11) Z

sin

25x2

dx.

1 ln p

1 6n(n2

4.11.

(1)

n2 + 3n

(2)

sin

(3)

; 1)

pn2

n=1

;

n=1 pn + 1

n=1

1 (

(4)

n + 2

(5)

(6)

;

1)n

;

+ 1) ln(2n)

n=1

;

n=1

; 7)2n;1

; 6x + 12

(7)

1 n2 arctg 2nx

(8)

(9)

n=1

(2n2

5n)4n

n=1

4n(n2 + 1)

;

e3x + e;3x

;

; 2 (11) Z

(10)

cos

4x2

dx.

4nn2

1 1 + cos n

1=3

4.12.

(1)

n2 + 2

(2)

arctg

(3)

+ 2)!

n=1

sin n

(4) 1 n5;n

(5)

(6)

. (7)

(3n

1) ln n

nx2+1 + 4

n=1

n=2

n=1

;

(8)

(9)

tg2nx. (10)

(11)

dx.

n=1 (3n + 1)2n

8 + 2x

n=1

;

p16 + x4

n cos2 n

4.13.

(1)

(2)

1 pn tg

;

(3)

+ 5

(n!)

n=1

n=2

;

n=1

(4)

2n + 3

(5)

(6)

(;1)n

n=1

n + 1

n=1

(2n

;

1) ln(n

+ 1)

n=1

n ln(2n)

(7)

1 2 + n

4n =x

(8)

(x ; 2)3n

(9)

1 ne;n= cos x.

n=1

(5n

;

8)3

n=1

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