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7

 

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# #' 7# *" (# * # * ( + * # .............................................................................................................

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0 # *+/0 0 #) !" *+ '* + * ( + * #.......................................................................................................

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4.

/( # / $ * ( + * #.............................................................................................................................

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5.

. * $+ ## * ( + * ' + # /( +.........................................................................................

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0 ( #/ *" (/6 '*/ * ( + * # * (, ) 1 ! * #+ */ + * * # (.

* * +

, + 1 * ) 1 ! * & */ 3 #0 ( & ( &. (# 0+ # * ( + * #, 3 ( 6 '

#7 * ( # *+ '* + # 2, ) 1 *+ '* + + *# *" (# * # * ( + * #.

 

( # 1. ! x1, x2, ..., xn, ..., %,

 

 

 

"e > 0 $n0 "n > n0, m > n0 |xn - xm| < e.

 

3 # + # #, ) + +* 36 * # " e > 0 ( * * ! 3 36 #/ ( #/ 1.

, * C > 0 -

$# *# +

)#*. * # e # # +* 2# ! 0 ) #/, Ce 2 # # +* + 0 2 !

2# ! 0 ) #/. .1 ( # 1 2 $0# + + * ( 6 +#(: x1, x2, ...,

xn,... %,

 

 

 

"e > 0 $n0 "n > n0, m > n0 |xn - xm| < Ce.

 

2. / / ' "

( ## 1 # */ * 6 * + + # (, ( ! 2 * ') *, ) ( # 1 + *# *" (# * #. ( 6 / #0+ * # #' 7# ( / * ( + * '.

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, %#/ 7 2. # #' 7# *" (# * # * ( + * #

1. ! x1, x2, ..., xn, ... , %.

. + 2( # ! 1 # " 3" (# & # ( * ) & * +#/. ) * ( 0 * + 3" (# * #. .* * ( + * x1, x2, ..., xn, ... *" (# */. .2, ) $ ( .

3 0 )#

limn®¥xn = l # 0 #7 ( #

 

(1)

"e > 0

$n0 "n > n0 |xn - l| < e.

* # +!3 6 (# m, m > n0, ( /

& 2 +! # */ + * +

(2)

|xm - l| < e.

#0 + * + (1) # (2) +!+ (#

 

|xn - xm| = |xn - l + l - xm| £ |xn - l| + |l - xm| < e + e = 2e

# 0 #7 ) !' 0

"e > 0 $n0 "n > n0, m > n0 |xn - xm| < 2e,

) ( 0!+ $ ( * * ( + * # x1, x2, ..., xn, ... .

. '( ( 0 * + ( * ) * # * +#/. .( 2#, ) * ( + * x1, x2, ..., xn, ... $ ( #

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xn0 +1 -e < xn < xn0 +1 + e,

* * ( + * xn0 +1 , xn0 +2 ,..., xn ,... & #) . ( + , 2 * +

{ x1 , x2 ,..., xn ,...} = { x1 , x2 ,..., xn0 } È{ xn0 +1 , xn0 +2 ,..., xn }

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& #) * +* ' * ( + * #.

 

 

 

 

 

 

. 2 # 3 %## 6 7 & #)

/ * ( + * # + " #' # #2 #' ( !. 3 0 )#

 

b =

lim

 

, a =

 

xn .

 

xn

lim

 

n®¥

 

n®¥

 

 

 

 

. *+ '* + + " & # #2 & ( + 0 #7

1."n b £ xn £ a;

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+ '* + $ ( * # 0 #7 + +#(

"e > 0 $k0 "k > k0 xnk - xmk < e.

(# # * ( + * + * *+ '* + 2 # )#

a - b = (a - xmk ) + ( xmk - xnk ) + ( xnk - b ) < e + e + e = 3e.

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"

3 + (# )

! 0

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* # + ) & #) *.

2. . 3 %## 4

* # & #)

* * ( + * # + * + *" (# *.

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, 3 & ( / 2 # 3 %## 6

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* * + , 2 2( */ + ( ( 0 * +.

* # ' * ' + *" 0 +#*# * ' 0 ! 4 % -' 7 **.

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n

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k =1

a2, ..., an, ... , S1, S2, ..., Sn, ... .

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n

S = lim Sn

= lim

ak

n®¥

n®¥

k =1

 

 

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an .

n=1

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b

 

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1

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/( bqn-1, |q| < 1, 3 ( 0!+ .

 

 

 

 

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.$# + # " ## /( + 3 %## 4 *" (# * # ' & #) '

* ( + * # # # #' 7# ( / * ( + * '.

 

 

 

 

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&( # &(, &( 1 * ( + * $ ( . # * + #, /( an *" (# */ &( #

&(, &(

 

 

 

 

 

 

n

m

 

 

 

 

"e > 0

$n0 "n > n0 , m > n0

ak - ak

< e.

 

 

k =1

k =1

 

& #)#+ / 36 * #, 2 *)#, ) m > n, * m = n + p ( / & & )#* p. .#

* & 7 ## ( * ! 3 0 + #/

n

m

 

n

n+ p

 

m

 

ak - ak

=

ak - ak

=

ak

< e.

k =1

k =1

 

k =1

k =1

 

k =1

 

 

 

 

 

 

 

 

 

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3. & an ,

 

 

 

n+ p

 

 

 

 

 

 

(3)

 

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ak

 

< e.

 

 

 

k =n+1

 

 

-#, # 2(, ) )#* e + + * + * +#/ (3) 2 0 # Ce * ! C > 0.

3 +! 2

3" (# # ( * ) * +# *" (# * # /(. * # * +# (3) * 3#, * " # */

3" (#,

3/0 * */ ( * ) ! * +# *" (# * # /(. * 6 * +# * 3 #,

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+ * + + (3) +! / * ( / +* " p ³ 1, #7 ( / ( & 0 ) #/ p = 1. 1 * ) * +#

(3) # 3 +#(

 

 

 

 

"e > 0 $n0 "n > n0 |an+1| < e,

# +! 2 * 6 * +, ) limn®¥an = 0.

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n®¥

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n+1

n

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k =1

k =1

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* ( * (, ) Sn £ Sn+1 ( / +* " !" n. - )#, * ( + * ) * !" * , # ! & +!

$# + # #' *" (# * # ' * ( + * # * + * + 6#' # #' *" (# * # /( + * #% ! # ) #.

5. !, an ³ 0 n. & an ,

S1, S2, ..., Sn, ... .

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1. .#0 %##2 #*(#*" &+)#* /(

 

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*" (# * #.

 

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*" (# */ # # . .* '7# #0 & # + * + #/ /+ / */ #0 2 %##, +! 2 !' + * ( 6 '

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1. ! n

 

|an| £ Cbn,

C > 0.

" bn |an|.

. & * # # 7# *" (# * /( bn 1 +#+ * +#

 

 

n+ p

 

"e > 0

$n0 "n > n0 "p ³1

bk

< e.

 

 

k =n+1

 

 

 

 

 

- ( / + * ( + * + 2 * #, ), * ( #0 * +#/ ! 1, bn ³ 0, n ³ 1.

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. *

 

n+ p

 

n+ p

 

ak

 

£ Cbk ,

 

 

#" (# * +#

k =n+1

 

k =n+1

 

 

 

 

 

 

 

 

n+ p

"e > 0

$n0 "n > n0 "p ³1 ak

 

< Ce,

 

k =n+1

) + *# *" (# * # /( |an| # ( 0!+ 1.

3 # + # #, ) + * + |an| £ Cbn ! 1 # / * ( / +* " n, )# / * n0. .1 * +#

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& ' #0 , &( #* 0 */ 0 # * + /(, #0+ * ( 0+ # #0 * + #/.$# & + * ( 6 ' .

2. ! bn ¹ 0, n ³ 1,

lim an = l, l ³ 0.

n®¥ bn

) l > 0, |an| |bn| . ) l = 0, |bn|

|an|.

. .* ( + *

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