Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://mph.cs.msu.su/stud/2014-MA-Grigorev.pdf Äàòà èçìåíåíèÿ: Mon Sep 1 17:38:24 2014 Äàòà èíäåêñèðîâàíèÿ: Sat Apr 9 23:15:34 2016 Êîäèðîâêà:

1.


1.1. . - , , , . 1.1.1. ( 1- ). R ­ . 1. ( -) x0 R = {x R : |x - x0 | < }, > 0 ­ . , x0 . , R . x0 = = {x R : |x| > K } , K > 0 ­ . (). x0 R, + = {x R : x0 < x < x0 + }, > 0 ­ .
3


+ x0 + x0 = + , {x R : x > K } , K > 0 ­ . x0 R , x0 = - . 2. f (x) g (x) D R . , G D f (x) = O g (x) , xG

(: "- g (x)"), M , x G |f (x)| M |g (x)| . 3. f (x) g (x) D, x0 ­ . , f (x) = o g (x) x x0 , (: "- g (x)"), x x0 (x) , f (x) = (x) · g (x) x x0 , D . 4. f (x) g (x) D, x0 ­ . x x0 , x0 f (x) = g (x) + o g (x) .
4


: f (x) g (x), x x0 (x D) . . 1) , 1 - 4, . 2) 2 G D , x0 , D ( 3 - 4). 3) 3 - 4 , x D. , , x0 , x x0 , . 4) ; , . 5) , 2 - 3, , . , , , x x0 f (x) = o g (x) , , o g (x) = f (x) . , o O , f (x) . 1.1.2. f (x) g (x) D, x0 ­ . x0 = ( x0 = + x0 = - ). .
5


1.
xx0

lim

f (x) = A, A = , g (x) x x0 .

f (x) = O g (x) ,

2.
xx0

lim

f (x) = 0, g (x)

f (x) = o g (x) ,

x x0 .

3.
xx0

lim

f (x) = 1, g (x)

f (x) g (x),

x x0 .

. 1) , 2 - 3, "" . . 2) 2 - 3 1. 3) 1 A = 0; , f (x) = O g (x) g (x) = O f (x) , x x0 , (.. f (x) g (x) x x0 ). : f (x) = O


g (x) ,

x x0 .

6


: f (x)

g (x) ,

x x0 .

4) : ) f (x) = O(1), x x0 , f (x) x0 ; ) f (x) = o(1), x x0 , f (x) ­ x0 . 1.1.3. 1. , a) sin x x, x c) sin x = O(x), x a) D = lim

0; b) sin x = o(x), x ; R. R , x0 = 0 , f (x) = sin x , g (x) = x .

f (x) sin x = lim = 1. x0 g (x) x0 x 3 sin x x, x 0. b) x0 = , sin x = o(x) , x , sin x lim = 0. x x c) - D = R . , | sin x| |x| x . 2 sin x = O(x), x R

2. , m, n a) xn = o xm , x 0 ; D ­ g (x) = xn , m < n . a) x0 = 0 , xn =
x0

N, m < n, b) x m = o x n , x + . x > -1, f (x) = xm , ox
m

,

x 0 , = 0.

lim

xn = lim x xm x0
7

n-m


b) , x 0 = + xm 1 lim = lim =0 x+ xn-m x+ xn x0 , D . 3. , x 0 : a) f (x) g (x) , f (x) = x 2 + sin 1 x , g (x) = x ;

b) f (x) = o(g (x)) , x2 , x Q , x, x Q, f (x) = g (x) = 0, x I, 0, x I. ( Q I ). a) 1 1 2 + sin 3, x 1 3 |x| |f (x)| = |x| · 2 + sin x |x| |f (x)| . , 2, f (x) = O g (x) g (x) = O f (x) , .. f (x) g (x) . , f ( x) 1 = 2 + sin g (x) x x 0 b) , f (x) = x · g (x) . (x) = x ­ x 0, 3 f (x) = o(g (x)) .
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f (x) , x0 g (x) , a = 0 , - -, 1 2, 2 3 . , a), lim f (x) , .. g (x) = x . 4. a) , x2 + 1 = O(x), x . x-2 , x2 + 1 lim = 1. x x(x - 2) , , x2 + 1 x, x x-2 b) x2 + 1 f (x) = sin 3x x 0 . x-2 g (x) = C x (C = 0) x2 + 1 sin 3x sin 3x f (x) 1 = lim · =- · lim . lim x0 x - 2 x0 g (x) C x 2C x0 x = 1 . f (x) 1 sin 3x 3 lim =- · lim =- =1 x0 g (x) 2C x0 x 2C
9


3 3 C = - , , f (x) - x , x 0 2 2 5. 1 ) . , x 0 : ex - 1 x , ln(1 + x) x , (1 + x)a - 1 a · x , a R , sin x x , x2 1 - cos x , 2 tg x x , arcsin x x , arctg x x . , x (x) : (x) 0 x x0 . 6. , 4 b) . x2 + 1 1 3 f (x) = sin 3x - sin 3x - x x-2 2 2 x 0 7. : x2 = o (f (x)) , x 0 , a) f (x) = x sin 2x ; b) f (x) = x cos(x2 ) ; 1 c) f (x) = 5 5 x ln(1 - x) ; d) f (x) = arctg(x2 ) ? x
10


: a) ; b) - d) . a)
x0

lim

x2 x 1 = lim = = 0, f (x) x0 sin 2x 2

. b) , x2 x lim = lim = 0. x0 f (x) x0 cos(x2 ) c) x2 x2 x2 lim = lim = lim = x0 f (x) x0 5 5 x ln(1 - x) x0 5 5 x (-x) =- .. . d) x2 x2 x2 lim = lim x · = lim x · 2 = 0 , x0 f (x) x0 arctg(x2 ) x0 x 8. : f (x) = O 3 a) f (x) = x ln 1 + 2 ; b) f (x) = x 1 c) f (x) = (2x + 1) · sin ; d) f (x) = x2 : a) - c) , d) .
11

1 5 lim x4 = 0 , 5 x0

1 x

, x ,

1 arctg(x2 ) ; 2 x 1 2+x · cos ? 3x - 10 x


a) ,
x

lim f (x) :

1 3 = lim x2 · ln 1 + 2 x x x = lim 3
y 0

=

y=

3 x2

=

ln(1 + y ) = 3. y

b) lim arctg(x2 ) =
x

, 2

1 1 = lim arctg(x2 ) = 0 , x x x x 1 1 f (x) = o , , f (x) = O , x . x x c) , lim f (x) :
x

lim f (x) :

1 1 = lim x (2x + 1) · sin x x x2 1 x2

= lim

x

x(2x + 1) = 2. x2

, sin d)
x

1 , x . x2 =

lim f (x) :

1 1 x(2 + x) = lim · cos x x 3x - 10 x x+2 1 · cos 3x - 10 x = = , 1 , 3

= lim x ·
x



x+2 1 · cos x 3x - 10 x lim

1 9. f (x) = x2 · sin x x = 0 . , , ­ ( x 0 ) :
12


(a) (b) (c) (d) (e) (f ) (g) (h)

x2 · sin x2 · sin x2 · sin x2 · sin x2 · sin x2 · sin x2 · sin x2 · sin

1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 x

= o(1) , = O(1) , = o(x) , = O(x) , = o(x2 ) , = O(x2 ) , x, x2 .

: (a) - (d), (f ). (e), (g ), (h). , (c) (d) ,
x0

lim

f ( x) 1 = lim x · sin = 0 . x0 x x

(f ) |f (x)| x2 . f (x) 1 (e) , = sin x2 x x 0 ( ) 1.1.4. 1. 1 - 3. 2. , , 1 - 3 .
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3. f (x), a) f (x) = o (x), x 0 ; b) f (x) = O(x), x 0 ; c) f (x) = o (x), x 5 ; d) f (x) = O(x), x 5 ; e) f (x) = o(x), x ; f ) f (x) = O(x), x ; 1 1 , x 0 ; h) f (x) = O , x 0; g ) f (x) = o x x 1 1 , x ; j ) f (x) = O , x . i) f (x) = o x x 4. a) c) e) 5. a) c) e) g) 6. 5. 1 7. f (x) = x2 · sin , x 9 , ­ , x . : x3 = o (f (x)) , x 0 , f (x) = x ; b) f (x) = x4 ; f (x) = x2 3 x ; d) f (x) = (x + 1)2 ; f (x) = (x arctg|x|)2 ; f ) f (x) = (x cos x)2 ? 1 : 2 = O (f (x)) , x , x 1 1 1 f (x) = 2 e|x| + 9 ; b) f (x) = · sin ; x x x 2 2 1 f (x) = x · sin ; d) f (x) = · cos ; x2 x x 3 1 1 f (x) = · arctg ; f ) f (x) = · arctg(x2 + 3) ; x x2 x 1-x f (x) = 3 ? x +4

14


1.2. 1.2.1. 1. f = f (x) D. x0 ­ , f (x) ( , , x0 ). x x0 : 1) o(f ) + o(f ) = o(f ) ; 2) O(f ) + O(f ) = O(f ) ; 3) O(f ) + o(f ) = O(f ) ; 4) o(o(f )) = o(f ) ; 5) O(O(f )) = O(f ) ; 6) O(o(f )) = o(f ) ; 7) o(O(f )) = o(f ) . , , : 1) o(f ) + o(f ) = o(f ). F (x) = g1 (x) + g2 (x) , g1 (x) = o(f (x)) , g2 (x) = o(f (x)) x x0 , g1 (x) + g2 (x) F ( x) = lim = 0, lim xx0 xx0 f (x) f (x) gi (x) = 0 (i = 1, 2) . lim xx0 f (x)
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, F (x) = o(f (x)), . 2) F (x) = g1 (x) + g2 (x) , |g1 (x)| M1 |f (x)| , |g2 (x)| M2 |f (x)| , M 1 , M2 > 0 ,

x x0 . x |F (x)| |g1 (x)| + |g2 (x)| M1 |f (x)| + M2 |f (x)| = = (M1 + M2 ) |f (x)| = M |f (x)| , . . 1) 1 (. 5) . 1.1.1). 2) , o(f ) - o(f ) = o(f ), O(f ) - O(f ) = O(f ) , .. . 1.2.2. , x x0 (x - x0 )n . , (x - x0 )n , n > 0 , n . x , xn , n > 0 , n . f (x) = Cn (x - x0 )n + C
16
n+1

M = M1 + M2 > 0 ,

(x - x0 )n

+1

+ ···+


+C

n+k

(x - x0 )n

+k

+ o((x - x0 )n+k ) ,

xx

0

( Cn = 0 ),

Cn (x - x0 )n , o((x - x0 )n+k ) ­ f (x) x x0 . . , , O (x - x0 )n+k . ( 1 ) . 2. D , x0 ­ , = (x) : (x) 0 x x0 , (x) ( , , x0 ). m , n N . : 1) o(n ) = o(m ) , m n ; 2) O(n ) = O(m ) , m n; m n; m n; ); );
m+n m+n

3) o(m ) + o(n ) = o(m ) , 4) O(m ) + O(n ) = O(m ) , 5) o(m ) · o(n ) = o( 6) O(m ) · O(n ) = O(

7) (o())n = o(n ) ; 8) (O())n = O(n ) ; 9) · o(n ) = o( o(n ) 11) = o(
17
n+1

); );

10) · O(n ) = O(
n- 1

n+1

), n > 1;


O(n ) 12) = O(

n-1

), n > 1.

. . 1.2.3. 1. : a) f (x) = 2x3 + x4 b) f (x) = 2x3 + x4 x c) f (x) = 3 2x + 5 x d) f (x) = 3 2x + 5 a) : 2x3 . x 0 f (x) b) : x4 . x f ( x) = 2 x3 + x4 = x x c) : . 5
4

x 0 ; x ; x 0 ; x . = 2x3 + x4 = 2x3 + o(x3 )

1+

2 x

= x4 + o x

4

x x x 0 f (x) = 3 = 2x + 5 5 x 2 x = 1 - x3 + o(x3 ) = + O(x4 ) 5 5 5 1 d) : . 2x2 1 0 , x x x x f (x) = 3 = 2x + 5 2x
18
3

2 1+ x 5

-1 3

5 1+ 2x

-1 3

=


5 1 1 1 1 1- 3 +o = 2 +O 2 3 2x 2x x 2x x5 c) - d) (1 + x)-1 = 1 - x + o(x) , x 0 . = 2. x 0 : a) f (x) = cos x ; b) f (x) = ex . x2 a) , x 0 cos x = 1 - + o(x2 ) . 2 2 sin2 x sin x 2 1 - cos x 2 2 lim = lim = lim = 1. x x2 x2
x0 2 x0

x , 2 x2 + o(x2 ) , x 0 . cos x = 1 - 2 , o(x3 ) , , , O(x4 ) , 1 - cos x b) , ex - 1 x , x 0 . , ex - 1 - x ex - 1 1 lim = lim =. x0 x0 x2 2x 2 x2 x x 0 e - 1 - x , .. 2 x2 x + o(x2 ) e =1+x+ 2 3. x 0 f (x) = sin x , .
19

2 2

x0

2


, sin x x , x 0 , .. sin x - x = o(x) . -2 sin sin x - x cos x - 1 lim = lim = lim x0 x0 x0 x2 2x 2x
2x 2

= 0.

, sin x - x = o(x2 ) .
x0

lim

sin x - x cos x - 1 1 = lim =- 3 2 x0 x 3x 6

( 2a) ) . , sin x - x = O x3 , x3 sin x = x - + o x3 6 1.2.4. 1 - 2 .5 1.1.3. x 0 ( ): x2 e =1+x+ + O(x3 ) ; 2
x

x2 ln(1 + x) = x - + O(x3 ) ; 2 a(a - 1) 2 (1 + x)a = 1 + ax + x + O(x3 ) ; 2 3 x + O x5 ; sin x = x - 6
20


x2 cos x = 1 - + O(x4 ) ; 2 tg x = x + O x3 ; arcsin x = x + O x arctg x = x + O x
3

; ;

3

x3 ex - e-x shx = =x+ + O x5 ; 2 6 x -x x2 e +e =1+ + O(x4 ) . chx = 2 2 1.2.5. {fn } {gn } , n N , , , n (.. -, , ). (), . . n : a n2 a) fn = ; a, b = 0 ; b) fn = 5 ; bn + c 2n + n3 - 1 1 1 c) fn = n2 + 1 - n2 - 3 ; d) fn = e n2 - ch ; n 1 2 n2 arctg n n arctg n ; f ) fn = . e) fn = n3 + 2 n3 + 2 1 n n , 1.2.4 .
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a) fn = = a bn

a a 1 = bn + c bn 1 + 1- c +O bn 1 n2

c bn

= =

a bn

1+

c bn 1 n2

-1

= .

a +O bn

a , n fn = bn + c a 1 , , 2 bn n n2 n2 1 b) fn = 5 = 5· 1 2n + n3 - 1 2n 1 + 2n2 -
n 1 2n

=
5

1 1 +o 2n3 n3

,

lim

1+

1 1 -5 2n2 2n n2

=1

c) fn = = 4 n

n2

+1- 1 1+
1 n2

n2 + 1 - n2 + 3 -3= = n2 + 1 + n2 - 3
3 n2

=

+

1- 1 + n2

1 2 +o n n 3 n2

,

n

lim

1+

1-

= 2.

: fn = n2 + 1 - n2 - 3 = n ·
22

1 1+ 2 n

1 2

3 - 1- 2 n

1 2

=


=n· 1+ =n·

1 +O 2n2 1 n4

1 n4 =

- 1- 2 +O n

3 +O 2n2 1 n3 ,

1 n4

=

2 +O n2
1

n 1 n4

d) fn = e n2 -ch

1 1 = 1+ 2 +O n n 1 +O 2n2 1 n4

1 1 - 1+ 2 +O n4 2n , n

=

=

e) fn =

n2 arctg n 1 arctg n 1 = = +o 2 n3 + 2 n 1 + n3 2n n n N , n 2

,

arctg n

1 1 1 n2 n + o n n2 arctg n f ) fn = = = n3 + 2 n3 + 2 1 1 n (1 + o(1)) = 2 +o =3 , n 2 n n2 n 1 + n3

1.2.6. 1. 2. 3. 1. 2. m , n N . , x : a) O(xm ) = O(xn ) , m n; m n;
m+n

b) O(xm ) + O(xn ) = O(xn ) , c) O(xm ) · O(xn ) = O(x
23

).


4. x 0 f (x) , : a) f (x) = ln(1 + x) ; b) f (x) = ax - 1 (a > 0) . 5. : x + sin x a) f (x) = 2 x 0 ; x +1 x + sin x b) f (x) = 2 x ; x +1 4 c) f (x) = x + 1 - x2 + 4x + 1 x 0 ; 4 d) f (x) = x + 1 - x2 + 4x + 1 x + ; 3 x · arctg x2 e) f (x) = x 0 ; 2 1+x 3 x · arctg x2 f ) f (x) = x . 1 + x2 6. x 0 f (x) : a) f (x) = x - sin |x| ; b) f (x) = tg2 (2x2 ) - 3x5 ; c) f (x) = 1 - 2x4 + x4 - 1 ; d) f (x) = x cos x + ln(1 - x) ; x2 x2 e) f (x) = + ln(cos x) ; f ) f (x) = e- 4 - cos x . 2 7. n : 2n + 1 2 a) fn = 2 -; b) fn = n + n ; n +1 n 1 1 1 5 + ; c) fn = d) fn = 1 - cos ; n n n 1 1 2 e) fn = ch - cos ; f ) fn = n - n - 1 · ln 1 + . n n n
24


1.3. () 1.3.1. D = {x : x > 0 } x + . 1. ln x x f (x) = o(g (x)) , x + , f (x) g (x) . . ( ) , . , ln ln x = o(ln x) , x + , ax = o ab
x

( > 0) (k > 0) (a > 1)

k

ax

, x + , a , b > 1 .

1 . ,
x+

lim

ln x =0 xk
25

k > 0 .


, 0 , > 0 , . D = {x : x > 0 } x0 = 0+ . 2. a
1 x

(0 < a < 1) x
k

(k > 0) 1 x ( < 0)

ln

f (x) = o(g (x)) , x 0+ , f (x) g (x) . 1. 1.3.2. , . 3. : ln n nk
26

( > 0) (k > 0)


an

(a > 1) n!

fn = o (gn ) , n , {fn } {gn } . . , {n!} . , 2 - 3, , . 1.3.3. . 1. f (x), g (x) , D , x0 . f (x) = o g (x) , x x0 , x0 , |f (x)| < |g (x)| . x x0 (x) , f (x) = (x) · g (x) x x0 . lim (x) = 0 ,
xx0

x0 |(x)| < 1 ,
27


, : 2. {fn } {gn } n fn = o(gn ) , N , n > N |fn | < |gn | . 1.3.4. 1. ´ x : a) f (x) = 106 x3 g (x) = 10-6 x4 ; b) f (x) = ln(x300 + 1) g (x) = 10 x ? a) A > 0 , f (x) < g (x) x > A . ,
x+

lim

f (x) 106 x3 1012 = lim = lim = 0, g (x) x+ 10-6 x4 x+ x

f (x) = o(g (x)) , x + , 1 . 1.3.3 b) x f (x) < g (x) , f (x) = o(g (x)) , x + ,
x+

lim

f (x) ln(x300 + 1) 300 ln x = lim = lim =0 10 10 x+ g (x) x+ x x

(. 1) 2. a) b) c) d) , n: fn = ln500 n gn = n0.001 ; fn = ln25 n gn = (0.95)n ; n fn = n100 gn = 100 n ; fn = (ln n)ln n gn = n10 ?
28


a) n0 , n > n0 ln500 n < n0.001 , fn = o(gn ) , n , 3. b) n fn , gn 0 , ln25 n > (0.95)n n > n0 . c) , lim n n = 1 . n 100 n lim fn = lim n = 1 . n n n gn = 100 n , , g n > f n n > n 0 . d) fn , : fn = e
ln n·ln ln n

=e

ln n ln ln n

= nln

ln n

.

{ln ln n} , n0 , ln ln n > 10 n > n0 . , fn > gn , n > n0 3. , n n < ln n . : 1) < 0 , ­ ; > 0 . 2) = 0 ,

1) < 0 . 0 . < 0 , ln- n = o (n- ) , n ( - > 0 , - > 0 ). ln- n < n
-

n < ln n n > n0 .
29


2) = 0 . , , n 3 ln n > 1 > 0 . 3) > 0 . ln n = o (n ) , n , > 0 , 0 . , ln n < n n > n0 , .. > 0 1.3.5. () 1.3.3 , . (. 2) . . 1. ­ , > 0 . N ( , ), n > N ln n 1 < . n1+ n1+ 2 ,
1 ln n < ln n < n 2 . n1+ n1+ 2 k = > 0 , , 2

ln n = o n
30

k

, n ,


2 ln n < nk n > N 2. ­ , > 0 . N = N ( , ) , n > N ln n 1 > . n1- n1- 2 . 3. k ­ . nk · e
-n

<

1 n2

n > N , N = N (k ) .

1.3.6. 2 . 1.3.1. 3 . 1.3.2. , x: f (x) = x arctg x g (x) = x ln x ; x2 b) f (x) = g (x) = x ln x ; x + 10 1 c) f (x) = x sin g (x) = 3 x ; x 1 ln x ? d) f (x) = g (x) = x x 4. , x 0 + 0 , .. x0 = 0 : 1 a) f (x) = x arctg x g (x) = x ln ; x
31

1. 2. 3. a)


x2 1 b) f (x) = g (x) = x ln ; x x + 10 1 c) f (x) = x sin g (x) = 3 x ; x+1 1 1 1 d) f (x) = g (x) = · ln ? x x x 5. , n : 100 a) n100 (1.001)n ; b) 100n n! ; c) nn n! ; d) ln k n k ln n , k N ; e) (ln ln n)ln n n2 ; f ) (ln n)ln ln n n ? 6. , a > 0 b n a) abn < n2 ; b) lnb n < an ; c) lnb n > an ? 7. 2 3 . 1.3.5.

1.4. , . . 1.2.4 . , -, x0 = 0 n N : 1. xn ex = 1 + x + · · · + + O xn+1 n! 2. x3 x2n-1 n-1 sin x = x - + · · · + (-1) + O x2n+1 3! (2n - 1)!
32


3.
2n x2 nx + · · · + (-1) +O x cos x = 1 - 2! (2n)! 2n+2

4.

x3 x2n-1 shx = x + + ··· + +O x 3! (2n - 1)! chx = 1 + x2 x2n + ··· + +O x 2! (2n)! x2 + · · · + (-1) 2

2n+1

5.

2n+2

6. ln(1 + x) = x - 7.
n-1

xn +O x n

n+1

(1 + x)a = 1 + a x +

a (a - 1) 2 x + ···+ 2! a (a - 1) · · · (a - n + 1) n + x + O xn+1 n!

33


2.


2.1. . 2.1.1. {an }. 1.


a1 + a2 + · · · + an + · · · =
n=1

an

(1)

. a1 , a2 , · · · an , · · · . 2.
n

sn = a1 + a2 + · · · + an =
k =1

ak

n- (1). , (1) {sn } . , {sn } , {sn } : a1 = s1 ; ak = sk - sk 3.
n=1 -1

(k > 1). an -



, {sn }, .. lim sn .
n

34




s = lim sn :
n n=1

an = s .

{sn } , (1) . 4.


rn =
k =n+1

a

k

n- . s , rn = s - sn . . , :


a)
n=1

q

n- 1

= 1 + q + q2 + · · · + qn + · · · .

: , |q | < 1 , .. . n , : 1 - qn sn = , q = 1. 1-q 1) |q | < 1, 1 1 - qn = , lim sn = lim n n 1 - q 1-q .. s =
35

1 . 1-q


2) |q | > 1 lim sn , .. n . 3) q = -1, 1 - 1 + 1 - 1 + ··· , s2n-1 = 1, s2n = 0 . , {sn } : 0; 1 . 4) q = 1 , 1 + 1 + 1 + ··· , sn = n ,
n

lim sn = , ..

b)
n=1

1 . n(n + 1)

: .

, an = 1 1 1 =- . n(n + 1) n n + 1

1 1 1 + + ··· + = 1·2 2·3 n(n + 1) 1111 1 1 1 = 1 - + - + + ··· + - =1- , 2233 n n+1 n+1 .. sn = s = lim
n

1-

1 n+1

=1

c)
n=1

1 . n(n + 1)(n + 2)
36

: .


ak = = sn = = 1 1 1 + + ··· + = 1·2·3 2·3·4 n(n + 1)(n + 2) 1 1 - + n n+1 1 . n+1 1 2 1 , 4 1 1 1 1 = - + = k (k + 1)(k + 2) 2k k + 1 2(k + 2) 1 2 1 1 - k k+1 + 1 2 1 1 - k+2 k+1 .

1 1111111 (1 - + - + - + - + · · · + 2 2322343 1 1 1 1 1 + - )= 1- + - n+2 n+1 2 2 n+2 ,
n

lim sn = lim

n

1 2

1-

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

=

1 2

1-

=

s =


d)
n=1

1 . n2

: .

1 1 1 1 < = - k2 k (k - 1) k - 1 k (k > 1) , (2)

: 1 1 sn = 1 + 2 + · · · + 2 < 2 n
37


111 1 1 1 + - + ··· + - = 2 - < 2 n . 223 n-1 n n 1 an = 2 > 0 , {sn } , n , ( ) <1+1- . 3 {sn } (. a) - c) ), ( d)). , ( ). , {sn } . {sn } , , . ( ), . 2.1.2. 1. 3 : (1) , s, > 0 N = N ( ), n > N : |s - sn | < . , , : rn = o(1) , n . 2 ( ). (1) ,
n

lim an = 0 .

(3)

38


n n

an = sn - sn-1 . , lim sn = s , , lim an = lim (sn - s
n n-1

) = lim sn - lim sn
n n

-1

= s-s = 0

. 1) :


lim an = 0 ,
n

an
n=1

. 2) (3) .


,
n=1

1 n

,

1 lim = 0 , .. n n . , : 1 1 n sn = 1 + + · · · + > = n . n n 2 , {sn }, , . , :


a)
n=0

q k = 1 + q + q 2 + · · · + q n + · · · , |q | 1 .

a) . 2.1.1. .
39


, |q | 1 , , lim q n = 0 . , q > 1 + , n q = 1 1, q -1


b)
n=1

2n3 . 3n3 + 1

, : 2n3 2 = =0 lim n 3n3 + 1 3


c)
n=1

sin n .

, . , , lim sin n = 0 . n sin(n + 1) = sin n cos 1 + cos n sin 1 , cos n sin 1 = sin(n + 1) - sin n cos 1 . , n , ,
n

lim (cos n · sin 1) = lim sin(n + 1) - cos 1 · lim sin n = 0 .
n n

sin 1 = 0 lim cos n = 0 . n ,
n

lim (sin2 n + cos2 n) = 1 ,
40


lim sin n = lim cos n = 0 . n n , lim sin n = 0
n

2.1.3. . (1) , > 0 N = N ( ), n > N , p N : |an
+1

+ · · · + an+p | < .

­ {sn }. , : {sn } c , > 0 N = N ( ), n > N , p N : |sn
n+ p +p

- sn | < .

|sn

+p

- sn | = |a

n+1

+ ··· + a

n+p

|=
k =n+1

ak

. .


a)
n=1

1 . n3 1 1 < 2, 3 k k k > 1,

(2), |a
n+1

+ ··· + a

n+p

|=

41


=

1 1 1 1 + ··· + < + ··· + < (n + 1)3 (n + p)3 (n + 1)2 (n + p)2 1 1 1 1 1 1 <- + ··· + - =- . n n+1 n+p-1 n+p n n+p , |a
n+1

+ ··· + a

n+p

|< 1

1 1 1 - << n n+p n

p N ,

n > N , N =


.

b)
n=1

1 n

( ).


n+ p

k =n+1

1 1 p n 1 1 + ··· + > = =, = k n+1 n + p n + p 2n 2

p = n. , = 1 > 0, N , n > N , p = n : 2


|an

+1

+ · · · + an+p | > ,

.. c)
n=1

cos(n!) . n3

|an
+1

+ · · · + an+p | =

cos(n + p)! cos(n + 1)! + ··· + (n + 1)3 (n + p)3
42


cos(n + 1)! cos(n + p)! 1 1 +· · ·+ +· · ·+ . 3 3 3 (n + 1) (n + p) (n + 1) (n + p)3 a). |an
+1

+ · · · + an+p | <

1 < n

n > N , p N ,




d)
n=1

2 + sin n n · (n + 1)

. + · · · + an+p | = 2 + sin(n + p) (n + p) · (n + p + 1) 1

|an = 2 + sin(n + 1)

+1

(n + 1) · (n + 2) 1

+ ··· + + ··· +

> (n + 1) · (n + 2) (n + p) · (n + p + 1) 1 1 > + ··· + = (n + 2) · (n + 2) (n + p + 1) · (n + p + 1) 1 1 p n n 1 = + ··· + > = > =, n+2 n + p + 1 n + p + 1 2n + 1 3n 3 p = n > 1. ( |2 + sin k | = 2 + sin k 1 b) ). , 1 > 0, N , n > N , p = n : |an 3 =
43
+1

+ · · · + an+p | > ,




e)
n=2

{ln n} . 3n

{ln n} ln n . , 0 {ln n} < 1 . |an <
+1

+ ··· + a + ··· + 1+

n+p

|= <

{ln(n + 1)} {ln(n + p)} + ··· + < 3n+1 3n+p 1 3n =
+1

1 3
n+1

1 3n+
p

+ ··· + 1 ·

1 3n

+p

1 3n+1 3n+1 1 - 1 3 , , > 0 N , = |an
+1

1

1 + ··· 3

+ ··· = 3n+p+1 1 = , p N . 2 · 3n

+

1

+ · · · + an+p | <

n > N , p N ,

f)
n=1

ln n . n5/2
2

n , ln n N , n > N < 1 (. . n1/2 1.3.5) . |an =
+1

ln n = o n1/

+ · · · + an+p | =

ln(n + 1) ln(n + p) + ··· + = (n + 1)5/2 (n + p)5/2

1 ln(n + p) 1 ln(n + 1) · + ··· + · < (n + 1)1/2 (n + 1)2 (n + p)1/2 (n + p)2 1 1 < + ··· + . (n + 1)2 (n + p)2
44


, (2) , a) , 2.1.4.


a)


n=1

an


n= m

an , m > 1,

.

, . a) , , ( ). b) c = 0 ­ .



n=1

an
n=1

c · an .

s ­ (1), c · s. , sn c · sn . , {sn } {c · sn } c = 0 . c = 0, .


c)
n=1

an


n=1

b

n

A B .
45




(an + bn ) A + B . An = a1 + a2 + · · · + an , Bn = b1 + b2 + · · · + bn . lim An = A , lim Bn = B . n n Cn = (a1 + b1 ) + · · · + (an + bn ) = = (a1 + a2 + · · · + an ) + (b1 + b2 + · · · + bn ) = An + Bn .
n n=1

lim Cn = lim (An + Bn ) = lim An + lim Bn = A + B
n n n

2.1.5. 1. , : 11 1 a) 1 - + - + ··· ; 3 9 27 1 1 1 b) + + + ··· ; 1·3 3·5 5·7 1 1 1 c) + + + ··· ; 1·3·5 3·5·7 5·7·9 d) n+2-2 n+1+ n ;
n=1

46


e) q sin + q 2 sin 2 + · · · + q n sin n + · · · (|q | < 1) ; f ) q cos + q 2 cos 2 + · · · + q n cos n + · · · (|q | < 1) . 2. ,
n

lim an = 0 .



an ?
n=1

3. , : 2n n n n a) ; b) ; c) ; n+2 n+1 ln(n + 1) n=1 n=1 n=1 d) 0, 001 + e) 0, 001 +
3

0, 001 + · · · +

n

0, 001 + · · · ;

cos + cos 2 + · · · + cos n + · · · ; n! f) . 2n n=1 4. , : cos (2n + 5) sin (n5 ) ; b) ; a) 2n n2 n=1 n=1


c)
n=1

1 + cos2 n ; n
1 sin n ; n



e)
n=1

f)
n=1

n · (n + 1) n=1 sin nx · arctg ( n + 3) n2

d)

arctg n

; x R .

5.
n=1

an

. , -

( 47




)
n=1

A

n

.

, . 2.2. . 2.2.1. . , an n n0 . , a) . 2.1.4 , , an n 1 . :


pn , pn 0
n=1

(1)

( , pn an ). ( an 0 , -1 ; , . 2.1.4 b) ). 1 ( ). (1) c ,
48


, .. C, sn C n . . (1) c. , {sn }, . . pn 0 , {sn } . , {sn } 2.2.2. 2 ( ).


p
n=1

n

(1)


n=1

pn .

(2)

n 0 pn pn , 1) (2) (1); 2) (1) (2). (1) (2) {sn } {sn } . pn pn sn sn . 1) (2) . 1 C, sn C n. sn C , , (1). 2) (1) . 1 , {sn } . , {sn }, (2) . , , pn 0 pn pn .. n, n n0 , n0 ­ .
49


. pn 0 , pn 0 pn = o(pn ) , n . (2) (1). pn = o(pn ) , n , 0 pn pn n n0 , 2 3 ( ). pn 0 , pn 0 pn = O (pn ) , n (.. pn pn , n ). (1),(2) . pn = O (pn ) , n n0 |pn | M1 |pn | pn , , M1 > 0, M2 > 0 , |pn | M2 |pn | n > n0 . , pn (3) (4)

0 pn M1 · pn , 0 pn M2 · pn .


2. , (2).
n=1

M1 · p

n



(3) (1). , (1), (4) 2 (2) 3 . 1 ( ). pn > 0 , pn > 0
n

lim

pn =L pn
50

(L = 0 , L = ) .


(1),(2) . pn lim = L , L = 0 , pn pn , n pn n ( L = 1) 1 2. pn pn , n . (1) (2) . . , . (. 2.4.4 h)), , 2 . . 2, 3, .


n=1 n=1

q n , |q | < 1 , ( ).

1 np

1.
n=1

1 np

(5)

51


a) p > 2 ; b) p < 1 ( (5) . 2.3.1, a)). 1 , p n

. 1 < p < 2 ,

1 1 1 p 1 . p 2 ; n2 np n 1 , (2.1.1, n2 n=1


d)),

1 (2.1.3, b)). n n=1 (5) p > 2 , p < 1 2 . :


a)
n=2

1 ; ln n 1 ; ln 1 + n



b)
n=1

2n · sin


1 ; 3n n · tg . n2 + 1 n 1 1 >. ln n n

c)
n=1

d)
n=3

a) 0 < ln n < n , n 2 ,


1 , n n=1 1 b) , sin n > 0 n 1 . 3 0 sin ( 0) 1 1 2 sin n 2n · n = 3 3
n

2 3

n

.

52





n=1

2 3

n

, 1 n

c) n 1 ln 1 + ln 1 +


> 0 .

1 n



1 , n . n

1 , n n=1 2 d) , . n n 1 ; · tg · = 2 . n n n2 + 1 n nn n , 1 n2 n=1 tg 2.2.3. 4 ( ).


1) (1) pn+1 q<1 pn
n=1

pn ,

pn > 0 , -

n n0 1 .

(6)

(1) . pn+1 1 , n n0 1 , (1) . 2) pn
53


1) , , , n0 = 1 . (6) p
n=1 n=1 n+1

q · pn q 2 · p q
n

n-1

· · · q n · p1 .

0 < q < 1 ,

p1 · q n , , (1) 2. pn+1 1 , .. pn+1 pn > 0 n , pn lim pn = 0 , (1) n 2) 5 ( ). (1) pn+1 = L . lim n pn 1) L < 1 ; 2) L > 1 . L < 1 , > 0 , L = 1 - 2 , .. L + = 1 - . N , n > N L- < 1 - 4 L > 1 1 + , .. L - pn+1
q < 1 (6). . , > 0 , L = = 1 .
54


(7) : pn+1 >L- =1 pn n > N ,

4 (1) pn+1 . 1) 4 q < 1 pn pn+1 < 1. pn 1 , , n n=1 pn+1 n = < 1, pn n+1 2) 5 L = 1 (1) . pn+1 , , lim =1 n pn 1 1 , , n n2 n=1 n=1 6 ( ). (1) pn ,



pn 0 . n=1 np 1) nq<1 . np 2) n1 .
55

n , n ,

(8) (9)


1) (8) , pn q n . (1),


q n , 0 q < 1 , .
n=1

2) (9) pn 1 n , .. (1) 7 ( ). (1) lim n pn = L , 1) L < 1 ; 2) L > 1 . 5. . 1) , (8) 6 n pn < 1 . 2) L = 1 . 8 ( ). lim n pn = L , 1) L < 1 (1) ; 2) L > 1 . , lim n pn , n , . 1) L < 1 . ­ , > 0 N , n > N n pn < L + . 1-L 1+L = , q = L + = < 1. 2 2 . 1) 6.
56
n n




2) L > 1 . L-1 . n = n , n N { k pk } kn , kn pkn > L - n . n > 1 L - n > L - 1 = 1 , pkn > 1 . lim pk = 0 , , (1)
k

. :


a)
n=1

xn , x > 0; n! 3 n! ; nn (-1) + 3 ; 2n+1
n n



b)
n=2

n ; (log2 n)n 4n + 1 2 ; 3n + 2 3 + (-1) 2n + 3n
n

c)
n=1

d)
n=1

1 · n7 n
10

nn

e)
n=1

f)
n=1

.

a)
n

lim

pn+1 xn+1 n! x = lim · n = lim = 0 < 1, n (n + 1)! x n n + 1 pn

, b) lim n n = 1 , n n n n lim pn = lim = 0 < 1, n n log2 n
57


c) pn+1 3n+1 (n + 1)! nn 3 · nn lim = lim = lim = n pn n (n + 1)n+1 3n n! n (n + 1)n = 3 lim
n

1 3 = > 1, 1 e (1 + n )n

, d) lim n 1 pn = lim 7 · n ( n n) 4n + 1 3n + 2
1 2

n

=

4 > 1. 3

, e) (-1)n+1 + 3 · 2n+1 pn+1 = n+2 = pn 2 · [(-1)n + 3] 1 , n = 2k + 1 , 1 , n = 2k . 4

1 (-1)n+1 + 3 =· = 2 (-1)n + 3

pn+1 , pn pn+1 lim n pn . 4 , . lim n pn = 1 · lim 2 n
n

n

(-1)n + 3 1 = < 1. 2 2

,
58


f ) , pn > 0 , n n10 3 + (-1)n n n lim pn = lim = n n n 2 + 3n n n10 3+1 3+1 = lim < 1. = n 3 n 2n 3· 1+
3

, . 4, 6 , ( , ­ ). , , : , . ( ) , (. e) ). 2.2.6.


1. an bn n N ,
n=1

b

n

.


n=1

an ?

2. , 1), 2) 6 7. 3. :
59


(1), pn > 0 , pn+1 lim < 1 , (1) . n pn


4. ,
n=1

an , an 0 ,


,
n=1

An .


( 5 . 2.1.5). 5. ,
n=1

pn ,

pn > 0 ,


n=1



pn p

n+1

. pn pn , pn > 0 ,



6.
n=1

+1


n=1

pn ?

7. , :


a)
n=1

1-

5

n ; n+1



b)
n=1

n · sinn x , |x| 1; n+2 n+3
n2

c)
n=1

n2 + 1 ; ln 2 n +3 n ; n + 4n 3
10

d)
n=1

;

e)
n=1

f)
n=1

sin8 (2n - 3) ; n (3n + 1)

60




g)
n=1

n! ; 1 · 4 · · · (3n - 2) (2n)! ; (n!)2 (2n)! (3n)! ; (n!) (4n)! j)



h)
n n=1

ln n + 2 cos(n!) ; n3 + ln3 n

i)
n=1

k)
n=1

arctg(n4 ) ; 2n + 1 + 3n - 1 =1 sin ( n + 1 - n) l) . 2n n=1

2.3. . () 2.3.1. - 1. f (x) 0 x > m , m ­ .


f (k ) = f (m) + f (m + 1) + · · ·
k =m

(1)

, n

{an }, an =
n m

f (x)dx , ..
+

lim
+

n

f (x)dx =
m m

f (x)dx

( :
m

f (x)dx ).

k - , k m + 1; x [k - 1, k ] ; f (k ) f (x) f (k - 1) .
61


f (x) , [k - 1 ; k ]
k k k

f (k )dx
k -1 k -1 k

f (x)dx
k -1

f (k - 1)dx ,

f (k )

f (x)dx f (k - 1) .
k -1

k m + 1 n :
m+1

f (m + 1)
m m+2

f (x)dx f (m) , f (x)dx f (m + 1) ,

f (m + 2)
m+1 n

......................... f (n)
n-1

f (x)dx f (n - 1) .

,
n n n- 1

f (k )
k =m+1 m

f (x)dx
k =m

f (k ) .

(1)
n

sn =
k =m

f (k ) ,

f (k ) 0 ,

sn - f (m) an s
n-1

.

(2)

an f (x) 0 , {an } . .
62


(1) . {sn } , , (2) , {an } . C , (2) sn an + f (m) . {an }, . {sn } , , 1 . 2.2.1 (1) . :


a)
n=1

1 ; np



b)
n=2

1 . n · ln n

a) p 0 , , . 1 p > 0 . f (x) = p x 1 . x , f (x) > 0 . + 1 dx p > 1 xp 1 p 1 . , :



n=1

1 p > 1 , p 1 np 1 x 2 . x · ln x

b) f (x) = f (x) > 0 , .

63



+ t = ln x dt = dx dt = t ln 2 2 x , , +

dx = x · ln x

2.3.2. . , . , , , . ­ . . pn > 0 n pn µ 1 =1+ +O . (3) pn+1 n n2 C C = 0, pn µ , n n . an = pn µ µ = n · pn . (3) (. 1/n . 1.4), n an+1 = an = 1+ n+1 n
µ

pn+1 · = pn

1 1+ n

µ

µ · 1+ +O n

1 n2

-1

=

1 1 µ µ 1 +O · 1- +O =1+O . n n2 n n2 n2 m n > m am+1 am+2 an an = · ··· . am am am+1 an-1
64


an ln = am ln an = ln am + sn ,
n-1 n- 1 k =m

ak+1 ln = ak

n- 1

ln
k =m

1+O

1 k2

.

(4)



sn =
k =m

ln



ak+1 . ak 1 k2

(4) , 1 k2

ln

1+O

=O

,

k .

s .
n

lim ln an = ln am + s ,

,
n

lim an = e

n

lim ln an

= e ln am + s = am · es = C > 0 .

,

C , n nµ 2 ( ). pn



n=1

pn ,

pn > 0 ,

(5)

µ pn =+ +O pn+1 n

1 n2

,

n ,

1) > 1 (5) , < 1 - ;
65


2) = 1 (5) , µ > 1 , µ 1 . , = 1 . ,
n

lim

pn+1 1 =, pn

5 . 2.2.3 > 1 (5) , < 1 - . = 1 . . ( 3 . 2.2.2) 1 nµ n=1 . 2 , (5) n µ pn =+ +O pn+1 n 1 n1+ , > 0.

.


1.
n=1

(2n - 1)!! 1 . (2n)!! 2n + 1

n : (2n - 1)!! · (2n + 2)!! · (2n + 3) pn = = pn+1 (2n)!! · (2n + 1) · (2n + 1)!! = (2n + 2)(2n + 3) = (2n + 1)2
66

1+

1 2n + 1

1+

2 2n + 1

=


1 1 1 3 1 1 +o 1+ +o =1+ +o 2n n n n 2n n , , = 1 , µ = 3/2 . = 1+ 1 1 = 2n + 1 2n = 1 1+ 2n
-1

,

=

1 2n

1-

1 +O 2n

1 n2

=

1 1 +O , n 2n n2 2. p , p (2n - 1)!! n! en ; b) : a) . (2n)!! nn+p n=1 n=1 pn a) n : = pn+1 = (2n - 1)!! · (2n + 2)!! (2n)!! · (2n + 1)!! =1+ p +O 2n + 1
p

=

2n + 2 2n + 1
2

p

=

1 1+ 2n + 1 1 n2 .

p

=

1 (2n + 1)

=1+

p +O 2n

2 = 1, µ = p/2, : , p > 2. (1 + x)p = 1 + px + O(x2 ) , 1 1 = +O 2n + 1 2n
67

x 0,

1 n2

,

n


b) 1 1 pn en · n! · (n + 1)n+1+p = · 1+ = n+p n+1 pn+1 n ·e · (n + 1)! e n 1 (n + p) · ln 1 + 1 n= = ·e e 1 1 p - +O n2 = 1 = e n 2n
n+p

= 1 n3

1 1 (n + p) - 2 +O 1 n 2n ·e e

=

p- 1 1 2 + +O , n . n n2 ( x 0 ) : x2 ln(1 + x) = x - + O(x3 ) 2
x 2

x =

1 ; n

p- 1 2 e = 1 + x + O(x ) x = (n ) . n , = 1 , µ = p - 1/2 , , p > 3/2 2.3.3. . {n!} . 3. n! A n · nn · e
-n

,

n

(6)

( A > 0 ­ ). n! pn = n -n ne n n! (n + 1)n+1 en 1 1 pn = = · 1+ = pn+1 (n + 1)! nn en+1 e n
68


-1 + n ln 1 + =e 1 +O 2n =e -

1 n 1 n2

-1 + n · =e =1-

1 1 - 2 +O n 2n

1 n3

=

1 1 +O , n . 2n n2 . 2.3.2 ( µ = -1/2 ), n! A pn = n -n -1/2 = A n , n , ne n A > 0 ­ . n! A n · nn · e-n , n . , (. ), A = 2 . . :


a)
n=1

n! ; nn



b)
n=3

ln(n!) . n2

a) , . (6) n n! A n · nn · e-n n · nn- n =A = pn . en nn nn , n0 , n > n0 n n - n > . , , n0 > e2 , 2 n · nn/2 n · en pn > A >A = A · n n > n0 . en en
69


, , b) , , pn > 0 : ln(n!) ln (A n · nn · e-n ) pn = = n2 n2 ln A + n + 1 · ln n - n n · ln n ln n 2 , n . = = n2 n2 n ln n ln n 1 , > n 3 ; n n n n=3 , 2.3.4. . 1. , , p : p k-1 k 1 a) k + 1 - k · ln · sin ; b) ; kp + 2 k k+1 k =1 k =2 k 2 k 2 + e-k c) logpk 1 + ; d) · arctg (k p ) . k 2k + 3
k =1 k =1

a) , , 1 sin > 0 , k 1 . k p 0 . k k = O (k 1/2 ) , p+2 k 2, p < 0, lim (k p + 2) = 3, p = 0. k
70


, . p > 0 . k k 1 p- 1 , kp + 2 k2 , k 1 1 1 1 · sin p- 1 · = p+ 1 , k . p+2 k k k2k k2 1 , p + > 1 p > 2 : , 1 p 2 b) , ak = k+1- k 1 . 2 1 p > , , 2

p

· ln

k-1 < 0. k+1 1 2p k

k k+1- k
p

=

1 k+1+



k

p



p/2

,

ln

k-1 2 = ln 1 - k+1 k+1 ak - 2p-
1

- 1 k .

2 2 - , k+1 k

1+

p 2

71




p , 1 + > 1 2
k =2 k =2

(-ak ) p > 0,

ak

p .

: , p > 0 , , p0 c) : p > 0, p = 1 . k k k ln 1 + k2 ln 1 + k2 2 = ak = logpk 1 + = , k ln(pk ) k ln p , ak ( ln p ). k k 2 1 1 ak , · = O k k ln p k2 , p > 0 , p = 1 d) , k 2 + e-k k , k . 2k + 3 2 0 arctg (k p ) = O (1) . , k , , , . 1 0 arctg (k p ) -p . k k1 1 · -p = O k , 2k k -(p+1) ak > 0 ,
72

p ak = O (k ) p < ak


, -1 - p > 1 p < -2 2. , :


a)
k =2

lnp k ; kk k ·e
p -k



b)
k =1

ln k ; kp


c)
k =1

;

d)
k =3

1 (ln k )ln
ln k

.

( , , (. . 1.3.5), .) a) p . (. 1.3.2), 4 lnp k = o( k ) k . k0 = k0 (p), lnp k < 4 k k > k0 , 4 lnp k k 1 0 < < = 5 k > k 0 . kk kk k4




5 , k4 k =1 p

1

b) p 1 ln k 1 1 > p , kp k k
73

k > 2,


. p > 1 . > 0 , p - > 1 . ln k = o(k ) k , ln k 1 1 = p · o(k ) = o kp k k p-


.



1

k p- k =1 p > 1

, p- > 1 ,

c) , k k ·e , lim k
p+2 ek p -k

1 <2 k

k

p+2




k

.

(7)

k

=

x= k k = x2

= lim

x

2(p+2)

x+

e

x

= 0 p ,

(. . 1.3.2). k p+2 k0 = k0 (p) , < 1 k > k0 , ek 1 (7). k2 k =1 d) a : a-1 = ln k k
ln ln k -1 k

,
(ln ln k )2

=e

ln ln k ·ln ln k

=e

.

74


, k (ln ln k )2 < ln k . , , (ln ln k )2 = o(ln k ) , k ,
k

lim

(ln ln k )2 (ln x)2 = { x = ln k } = lim = x+ ln k x = lim 2 ln x ·
x+

1 = 0. x

1 1 2 = e-(ln ln k) > e- ln k = k > k 0 , ln k ln ln k k ak = 2.3.5. . 1. f (x)
+

x 1 , f (x) 0 ,
m

f (x)dx

-

.
n=1

f (n) ?

. , : 1 1 f (x) = 1 - n2 · |x - n| , x n - 2 ; n + 2 , n 2 ; n n f (x) = 0 x 1 .


2.
n=1

an .

75


) an = o ? ) an = O ? ) an = O ?


1 n 1 n 1 n

, , ,

n , n , n , -

an 3. , a > 0, nb · (ln n)c n=3 , a, b, c a) ; b) . 4. , :


a)
k =1

k · (2k - 1)!! ; (2k )!! k · e-pk ; 3 ln k k! · k ; 5 · 6 · · · (k + 5)
p



b)
k =1

2 · 5 · · · (3k - 1) ; 3k · k ! sin 1 k · ln k (2k )! . (2k · k !)2 ;

c)
k =2

d)
k =2

k =1

e)
k =1

f)

5. , , p : 5 1 k k+1- k-1 a) · tg ; b) ; 2k p + 1 k kp
k =1

c)
k =1

ln

k +1 ; kp
76

p

k =2

d)
k =1

ch - cos k k

p

.


6. , :


a)
k =2

lnp k ; k lnp k e
3 k



b)
k =2

kp ; ln2 k 1 (ln ln k )ln
k

c)
k =1

;

d)
k =3

.

77