Сборник задач по высшей математике 2 том

a) F(P) = 4

P

j

6) F(P) = e-P + e-2p

P -

4r + 4

p2 + 6p + 10

4. PeIIIHTh CHCTeMY .n;H<p<pepeHIIHaJIhHhIX ypaBHeHHfi

X' + X -

y = 0,

{

x(O) = y(O) = O.

3. HafiTH OpHrHHaJIhI no CJIe~IOrn;HM H306proKeHH5IM:

a) F(p) -

2

.

6) F(P) -

-p

P

--::-=pe'----

- (P2

+

4)2 '

-

p2 + 8p

+ 20

4. PeIIIHTh .n;H<p<pepeHIIHaJIhHOe ypaBHeHHe x" + 4x = et ,

x(O) = x' (0) = O.

t

5. PeIIIHTh HHTerpaJIhHOe ypaBHeHHe x(t) - t = Jsin(t -

u) . x(u) duo

o

f(t)

f(t)

1

1

3

t

t

Puc. 122

Puc.

129

1.1.2. Sn = 0 - IIpH qeTHOM nj Sn = 1 -

IIpH HeqeTHOM nj

lim Sn He

n--+oo

cym;ecTByeTj pH)l; paCXO)l;HTCH. 1.1.3. Sn = n 2 j

lim Sn = +OOj pH)l; pacXO)l;HTCH.

1.1.4. Sn = {2k +

IIpH n = 2k +

n--+oo

Sn = OOj pH)l;

2

Ij -2k IIpH n = 2k}j

lim

n--+oo

paCXO)l;HTCH. 1.1.5. Sn = 2n - Ij

lim Sn = +OOj pH)l; pacXO)l;HTCH.

1.1.6. Sn = ~ (1 - 2~)j

n--+oo

nl~rr;,Sn = ~j pH)l; CXO)l;HTCH.

1.1.7. Sn = ~ (1 -

2n ~ 1) j 1~rr;,Sn = ~j pH)l; CXO)l;HTCH. 1.1.8. Sn = In(n + l)j

lim Sn = +OOj pH)l; paCXO)l;HTCH. 1.1.10.

lim

1

an = -2 j pH)l; paCXO)l;HTCH.

n~~

n~oo

1.1.11.

lim an = O. 1.1.12. lim

an = OOj pH)l; pacXO)l;HTCH. 1.1.13.

lim an = 1!:2 j

n-+oo

n-+oo

n-+oo

PM paCXO)l;HTCH. 1.1.14.

lim an = O. 1.1.15.

lim an = OOj pH)l; pacXO)l;HTCH.

n-+oo

n-+oo

lim an = e3 j PM

1.1.16.

lim an He cym;ecTByeTj pH)l; paCXO)l;HTCH. 1.1.17.

n-+oo

n-+oo

paCXO)l;HTCH. 1.1.19.

~+1

(5)n

CXO)l;HTCHj ~. 1.1.20. PaCXO)l;HTCHj

2 .

1.1.21.

PacXO)l;HTCHj

1

CXO)l;HTCHj

1

PaCXO)l;HTCHj

1

r,;::. 1.1.22.

2·1.1.24.

n·

yn

n

1.1.25.

1

1.1.26. PaCXO)l;HTCHj

n

1

CXO)l;HTCHj

-

n 2 •

3·1.1.27.

PaCXO)l;HTCHj

n·

1.1.28.

1

1

CXO)l;HTCHj

1

PacXO)l;HTCHj -1 . 1.1.29.

CXO)l;HTCHj 7· 1.1.30.

2·

n2

n6

n

1.1.31.

PaCXO)l;HTCHj

1

1

8

n· 1.1.32. CXO)l;HTCHj - 3 . 1.1.33. PaCXO)l;HTCHj

n·

2)n

n2

1.1.34.

CXO)l;HTCHj

. 1.1.36.

PaCXO)l;HTCHj 2. 1.1.37.

CXO)l;HTCHj

1

(5

3·

1.1.38. CXO)l;HTCHj O. 1.1.39. CXO)l;HTCHj ~. 1.1.40. PacXO)l;HTCHj ~.

1.1.41. PaCXO)l;HTCHj

~. 1.1.42. CXO)l;HTCHj ~. 1.1.44. PaCXO)l;HTCHj e.

1.1.45.

1/21

CXO)l;HTCHj

2· 1.1.46. CXO)l;HTCHj

V3· 1.1.47.

CXO)l;HTCHj e2 •

1.1.52. PaCXO)l;HTCHj In In(x + l)j +00. 1.1.53. CXO)l;HTCHj

1

j _1_.

In(x + 1)

In 2

1.1.54.

IIpH P > 1

1

1

x p - 1

CXO)l;HTCHj

1 j --1· IIpH P < 1 pacXO)l;HTCHj -1-- j

(l-p)x P -

p-

-p

3)n

"1.

rrpH3HaK cpaBHeHHH; (2 . 1.1.58. PacXO,n;HTCH; 2-it rrpH3HaK CpaBHeHHH;

n3

3

2

8

HHTerpaJIbHblit rrpH3HaK; 2(lnx)3; +00. 1.1.63. CXO,n;HTCH; rrpH3HaK KorrIH;

125.

rrpH3HaK ,naJIaM6epa; 3. 1.1.71. Sn = n;

lim

Sn = +00; PH,n; pacxo,n;HTCH.

n-+oo

1.1. 72. Sn = _1 +2 n . n;

lim Sn = -00; pH,n; pacxo,n;HTCH. 1.1.73. Sn = (_I)n . n;

n-+oo

(1- 5~);

lim Sn = -4 ;

lim Sn = 00; PH,n; pacxo,n;HTCH. 1.1.74. Sn = -4

PH,n;

n--+oo

5

n--+oo

5

lim Sn = +00;

CXO,n;HTCH. 1.1.75. Sn = {5k rrpH n = 2k;

5k + 2 rrpH n = 2k + I};

n-+oo

pH,n; pacxo,n;HTCH. 1.1.76. Sn = 1 -

1

2;

lim Sn = 1; PH,n; CXO,n;HTCH.

(n +

1)

n-+oo

1.1.77.

lim

1

lim

an = In Q.2; PH,n; pacxo,n;HTCH.

an = -3 ; PM pacXO,n;HTCH. 1.1. 78.

n-+oo

n-+oo

1.1. 79.

lim

an = 1; PH,n; pacXO,n;HTCH. 1.1.80.

lim

an = !!:2; PH,n; pacxo,n;HTCH.

n--+oo

n-too

lim an = O.

1.1.81.

lim

an He CYIIIecTByeTj pH,n; pacxo,n;HTCH. 1.1.82.

n-too

n-too

1.1.83.

lim

an = 2; pH,n; pacxo,n;HTCH. 1.1.84.

lim

an = O. 1.1.85. PacXO,n;lITCH;

n-+oo

n-too

(2)n

1

1

1

1.1.88. CXO,n;HTCH;

2n· 1.1.86. CXO,n;HTCH; 3n · 1.1.87. PacXO,n;HTCH; n;.

5" .

1.1.89.

1

2

1

PacXO,n;HTCH; n;. 1.1.90. Pacxo,n;HTCH;

3· 1.1.91. CXO,n;HTCH; - n

2 •

1.1.92.

1

3

1

CXO,n;HTCH; "2. 1.1.93. Pacxo,n;HTCH;

r,;;. 1.1.94. Pacxo,n;HTCH;

n;.

n

2yn

1.1.95.

Pacxo,n;HTCH; n;.

1.1.96. PacXO,n;HTCH;

n;. 1.1.97. Pacxo,n;HTCH; n;.

1.1.98.

4~2

(5)n

1

CXO,n;HTCH; 7·1.1.99. Pacxo,n;HTCH;

2

. 1.1.100. CXO,n;HTCH; v'5.

1.1.101. PacXO,n;HTCH; ~. 1.1.102. CXO,n;HTCH; 0.1.1.103. Pacxo,n;HTCH; +00.

3

3

1.1.104. PacXO,n;HTCH; e. 1.1.105. Pacxo,n;HTCH; +00. 1.1.106. CXO,n;lITCH; 4".

1

e

.

1

1

.1.107. PacXO,n;HTCH; 2·1.1.108.

CXO,n;HTCH;

3·1.1.109.

CXO,n;HTCH; g.

1

2

1.1.110. CXO,n;HTCH; O. 1.1.111. CXO,n;HTCH; e· 1.1.112. CXO,n;HTCH;

3.

rrpHMeHHM. 6)

3

an

= -bn = 2 +

(_I)n. 1.1.142. TIpH3HaK

CXO.n.HTCH; 4. 1.1.141.

1

.

1

.

-2 . 1.2.11. 2-iI: rrpH3HaK CpaBHeHHH,

2". 1.2.12. TIpH3HaK )J;arraM6epa, O.

n

n

1

1

1

1.2.13. HHTerparrbHhlil: rrpH3Hax; -In n;

In 2·

1.2.14. TIpH3HaK )J;arraM6epa; "2.

1.2.15. Heo6xo.n.HMhlil: rrpH3HaK; +00. 1.2.16.

4

TIpH3HaK )J;arraM6epa; 3.

1.2.17. Heo6xo.n.HMhlil: rrpH3Hax;

~. 1.2.18. TIpH3HaK KOillH; ~. 1.2.19. CXO.n.HTCJ

a6cOJIIOTHO; 2-iI: rrpH3Hax CpaBHeHHH;

en~l. 1.2.20. Pacxo.n.HTCH; Heo6xo.n.HMhlil:

1

1

rrpH3HaK ,1l;anaM6epa; 2'1.2.31. PaCXO,l1;HTCH; Heo6xo,l1;HMblit rrpH3HaK;

2'

12 +il

J5

1.2.32. CXO,l1;HTCH a6cOJIIOTHO; rrpH3HaK KOillH; --3-

= 3'1.2.33. CXO,l1;HTCH

a6COJIIOTHO; rrpH3HaK ,1l;anaM6epa;

-I_1_'1= ~, 1.2.34. CXO,l1;HTCH YCJIOBHO; 2-it

2 +,

v5

2

2

13 - il

v'TIi

13il

= 3'1.2.36. PaCXO,l1;HTCH; rrpH3HaK KOillH; - 2 -

= -2-,1.2.37. CXO,l1;HTCH

1

1

n

vn

3

2

1.2.39. llHTerpanbHblit rrpH3HaK; 2(1n n +

2) 3, 1.2.40. I-it rrpH3HaK CpaBHeHHH;

1

1

1

.,fo'1.2.41. 2-it rrpH3HaK cpaBHeHHH; n'

1.2.42. IIpH3HaK ,1l;anaM6epa; 2'

1.2.43. 2-it rrpH3HaK CXO,l1;HMOCTH;

~, 1.2.44. IIpH3HaK KOillH; ~, 1.2.45. I-it

n2

1

1

In 2'1.2.50. PaCXO,l1;HTCH; Heo6xo,l1;HMblit rrpH3HaK; 2'1.2.51. CXO,l1;HTCH

a6cOJIIOTHO; rrpH3HaK ,1l;anaM6epa;

~, 1.2.52. CXO,l1;HTCH a6COJIIOTHO; 2-it rrpH3HaK

CpaBHeHHH;

1

. c'1.2.53. PaCXO,l1;HTCH; rrpH3HaK KOillH; +00, 1.2.54. CXO,l1;HTCH

3nvn

*'1.2.55. CXO,l1;HTCH a6COJIIOTHO; I-it rrpH3HaK

YCJIOBHO; 2-it rrpH3HaK CpaBHeHHH;

CpaBHeHHH;

3~' 1.2.56. CXO,l1;HTCH a6COJIIOTHO; HHTerpanbHblit rrpH3HaK;

(2+Inn)-2 1

1

12 +il

J5

npH3HaK KOillH; 3'1.2.60. PaCXO,l1;HTCH; rrpH3HaK ,1l;anaM6epa;

--3-

= 3'

11 + il

..J2

a6COJIIOTHO; rrpH3HaK ,1l;anaM6epa; --3-

= 3'1.2.63. CXO,l1;HTCH a6COJIIOTHO;

n2

CpaBHeHHH; k, 1.2.65. CXO,l1;HTCH a6COJIIOTHO; rrpH3HaK KOillH;

~,

YC.JlOBHO. 1.2.75. Y7Ca3a~ue.

1 ~

n+l

1.2.76.

Y7Ca3a~ue. 2an bn ~ a; + b;.

- n - ~ 2.

+00. 1.3.12. (2,4); (3 -

X)2. TIPH X = 2 H X

= 4 pH,l1; pacXO,l1;HTCH; 2-it npH3HaK

1

.

= -1 pH,l1; CXO,l1;HTCH YCJIOBHO; 2-it

cpaBHeHHH; fo' 1.3.13. [-1,1); IxI'.TIPH x

1.3.20. (2 - V2,2 + V2);

(x -

2?

2

. IIpH X = 2 - V2 H npH x = 2 + V2 pH,l1;

Ix-31

x

= 0 pH,l1; paCXO,l1;HTCH; Heo6xo,l1;HMbIit npH3HaK; lim

an He

--3-'TIPH

n-+CXl

cyru;ecTByeT. TIPH x = 6 pH,l1; pacXO,l1;HTCH; Heo6xo,l1;HMblit npH3HaK;

1

3'

1

2-it npH3HaK CpaBHeHHH; _1_; npH3HaK JIeit6HHu;a. TIPH x = -2 pH,l1; paCXO,l1;HTCH;

vn

x 2

Ix+21 2

= -2 - .;3 H npH x

= -2 +.;3 pH)l; paCXO)l;HTCHj Heo6xo)l;HMbIit

--3--·IIpH x

1

1.3.33. (-2, O)j npH3HaK KOnIHj Ix + 11- IIpH X = -2 PM

npH3HaKj =F .;3.

paCXO)l;HTCHj Heo6xo)l;HMblit npH3HaKj

lim an He cyru;eCTByeT. IIpH x = 0 pH)l;

n-too

1

.

1

nj npH3HaK

1

+

1

1

-

1

12

00

sin(2k -

l)x

.

1.4.2. -2

-2 cos 2x.

1.4.3. -2

-2 cos 2x. 1.4.5. --;r

E

k

-

1

k=l

2

00

(_I)n sin nx

00

(-It+l sin nx

1.4.7. 1 + 4 E

n

. 1.4.8. -3 + E

n

.

n=l

n=l

1.4.10. 1 -

16 00 sin(2k - l)x

5

IS

00

sin(2k -

l)x

-;r

E

2k

1

. 1.4.11. --2

+ -;r E

2k

1

.

k=l

-

k=l

-

11"2

+ 4

00

nCosnx

11"

4

00

cos(2k-l)x

11"2

1.4.12. -3

E (-1)

- 2 - · 1.4.13. -2 -

7i'

E

(2k -

2

. 1.4.14. -6 .

n=l

n

k=l

1)

11"2

11"

+

1

00

cos(2k -

l)x

1.4.15. -S . 1.4.16. 1 -

-S

7i'

E

(2k -

2

•

k=l

1)

1417

2sin1l"a (sinx

_

2sin2x + 3sin3x _

)

••

•

11"

1 _ a2

22 _

a 2

32 _

a2

• ••

•

11"

2

00

cos(2k -

l)x

00 1

n.

11"2

1.4.18. -4 - 7r

E

(2k -

1)

2

+ E n(-I)

smnx. 1.4.19. -S .

k=l

n=l

1.4.21. a) 2 f: (_I)n+lsinnnx; 6)

f: sin22nx.

n=l

n=l

n

1.4.23. a )

11"

4

~ cos(2k -

l)x . )

11"

2 ~ cos(2k - l)x.

--2

+ 7i'

L..J

(2k -

2

,6

-1 + -4 -

7i'

L..J

(2k -

2

'

k=l

1)

k=l

1)

2

00

2

+

00

0) _ZI... - 4 E (_ltCOS~x; r)

3 + ZI...

2 E (_ltcos~x.

3

n=l

n

6

n=l

n

4

00 (_I)n+lsin n~x

1 00 (-It+lsin2n1l"x

1.4.25. 7i'

E

n

. 1.4.26. 7i' E

n

.

n=l

n=l

(2k -

1)1I"x

1.4.27. 1 -

S

00

cos

2

1

-

1

00

cos4(2k-l)1I"x

2"

E

(2k -

2·1.4.28. -S

2" E

(2k -

2·

11"

k=l

1)

11"

k=l

1)

12

(

l)n

n1l"x

1

2

(_I)n cos 2n1l"x

00

-

cos -3-

+

00

1.4.29. 6 + 2"

E

n

2

•

1.4.30. -6

2"

E

2

•

11"

n=l

a + b

11"

n=l

n

S ~ sin(2k - l)x

2(a -

b)

~ sin(2k -

l)x

1.4.31. 7 -

7i' k~l

2k _ 1

. 1.4.32. - 2 - -

11"

k~l

2k -

1 .

1 4 33

•

1.

~ cos(2k -1)x

1434

2sin1l"a

(...!... + acosx _

a cos 2x +

)

••

1I"L..J

2·

•••

11"

2

a

2

2

2 ....

k=l

(2k - 1)

1 - a

2

- a

1435

•

Sa (cosx + cos3x + cos5x +

)

••

11"2

12

32

52

. .• .

11"

2

00

cos(2k-l)x

+

00

nsinnx

11"2

11"2.

1.4.36. -4 - 7i'

E

(2k -

2

E (-1)

- n - · 1.4.37. -S . 1.4.38. a)

-S ;

k=l

1)

n=l

00 .

00 (-l)n sin n1l"x

6) l1.4.39. 2 n~/

_l)n+l smnnx. 1.4.40. ~ n~l

n

1

1

4l

cos

(2k -

1)1I"x

212

4l

(

n

n1l"x

00

1

+

00

-1)

cos -l-

1.4.41. -2 - 2" E

(2k -

2· 1.4.42. -3

2"

E

2

11"

k=l

1)

11"

n=l

n

n=l

n

+ 1

n=l

n + 1

1.4.44. a) *E(_I)n+1 (~ +

~ ((_I)n -1)) sinnx;

n=l

n

4

<Xl

(2k -

1) sin(2k - l)x

rrpH qeTHOM a,

6) cos ax = -1i'

E

2

2

k=l

a - (2k - 1)

cos ax = -:# E 2: sin 2k~ rrpH HeqeTHOM a.

k=l a

- (2k)

.

4a

<Xl

cos(2k -

l)x

qeTHOe;

1.4.45. sm ax

= 1r

E

2

2 ' eCJIH a -

k=la -(2k-l)

.

2

4a

~

cos2kx

2 ' eC.JlH a - HeqeTHoe.

sm ax = rra +

1r

L..J

2

(2k)

k=l a

-

2.1.3.

a)

Her, 6) ,l1;a; B) HeT; r) ,l1;a. 2.1.5. a) y = x 2 + x-I; 6) y = 1 - i e- 3",.

2.1. 7.

a)

y = 3x + C; 6) y = 21n Ixl + C. 2.1.8. a) xy' + y = 0;

2.1.18. x 2 + y2 = In Cx2. 2.1.19. Y + In Iyl = sin x - x cos x + C.

2.1.20. Y = In(1 + Ce-"'). 2.1.21. y = __2_ . 2.1.23. Y = (x + 1)2.

2

C+x

2.1.24. y = (4x + 2)

2

.2.1.25.

x+y

y = 1. 2.1.26. - 1 -- = -3. 2.1.28. 200 M.

-xy

2.1.45. ~2

-

e-Y(y + 1) = C. 2.1.46. x + y = In(C(x + l)(y + 1)), Y = -1.

2.1.47. y =

2

2:& + sin 2:&

5 + Ce-"'. 2.1.48. v = Ce2t . 2.1.49. Y = Ce

4