Добавил:
Upload Опубликованный материал нарушает ваши авторские права? Сообщите нам.
Вуз: Предмет: Файл:

Матан Лекции

.PDF
Скачиваний:
19
Добавлен:
09.06.2015
Размер:
3.53 Mб
Скачать

com.neevia.http://www version trial Converter Personal Neevia by Created

, %#/ 9 1. 3* / # * + / *" (# * )#* + & /(

a = a+ - a-, |a| = a+ + a-.

1. ) an , .-

, an+ an-.

. . * 6 * + + 1 +!* 0 ! (+ + 2( #/. )

* + & #0 #", # "

3" (# & # ( * ) & * +#/ 3* ' *" (# * # /(.

 

 

 

 

 

. * /( an 3*

*" (# */. - )#, *" (# */ /( |an|. 0

+ * +

 

 

0 £ a

+ £ |a |

 

# 0 £ a

n

- £ |a

|

 

 

 

n

 

n

 

 

 

 

 

n

 

#0 2 %## * ( *" (# * /( + a

+

# a

 

-, ) ( 0!+ 3" (# * * +#/ + + ' ) * #

! 1. . #, )

 

 

n

 

n

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

n

n

 

 

n

 

 

n

 

 

 

n

n

(1)

ak

= ak+ -

ak-

 

ak

 

 

= ak+ + ak-

 

 

# 1

k =1

k =1

 

 

k =1

 

 

k =1

 

 

 

k =1

k =1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

¥

¥

 

 

¥

 

 

¥

 

¥

¥

(2)

an

= an+ -

an-

 

 

an

 

 

= an+ + an-.

 

 

 

 

n=1

n=1

 

 

n=1

 

 

n=1

 

 

 

n=1

n=1

3 , * /(! an+ # an- *" (/ */. &( * + ( #+! * 7 #/ (1) 2( ) * ! # * # ** #+ !" /( +, #0 !" * ( *" (# * /( + |an| # an # * + ( #+ * $ (2). 8 ( 0!+ ( * ) * * +#/ + + ' ) * # ! 1.

( + , # # / + + # # 3* *" (# * 7 & /(, ! ( 0 # + + 2( # # 0 + 7# # ( 0 * + ! 1.

(# 0+ # *" (# * #, #+ ' 3* ' *" (# * #.

( # 2. & an , , .

80

com.neevia.http://www version trial Converter Personal Neevia by Created

, %#/ 9 2. .3 0 + # 3 /

1 # *+ * 3* + #+, #*" (/6 #0 + ' $! + (1). * # * (+ " * ( + * ' *" (# */, #3 3 * (!+ ! * ( + * # *" (/ */, #3 3 # *" (/ */. 8

#+ (# * + 2( #: a

n

, a

+

a

-

.

 

 

 

n

 

n

 

2. ( *

 

 

 

 

 

 

 

#0 " *" (# * # /( + 0 * ( 6 / $ ,

0!+ / 3 0 + # 3 /.

n

 

 

 

 

 

 

 

1. ! Bn = bk , n ³1. " n ³ 1 %

 

 

 

 

k =1

 

 

 

 

 

 

 

 

 

n

n-1

 

 

 

 

 

 

ak bk = an Bn - (ak +1 - ak ) Bk .

 

 

 

 

 

 

k =1

k =1

 

 

 

 

. .# # / + + # # ) +#( ! $!

b1 = B1, b2 = B2 - B1, ..., bn = Bn - Bn-1,

)#

n

ak bk = a1B1 + a2 (B2 - B1 ) +K+ an (Bn - Bn-1 ) =

k =1

B1(a1 - a2) + B2(a2 - a3) + ... + Bn-1(an-1 - an) + Bnan =

n-1

Bk (ak - ak +1 ) + Bn an ,

k =1

) 1 +#+ $ + 1 # 0 )#+ ( 0 * +.

81

com.neevia.http://www version trial Converter Personal Neevia by Created

%#/, 9 3. .#0 #"# #*(#*" &+)#* /(

3. ( 1

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

3* , # * + *" (# *. ( 6#' #0 # ( 2# # #".

2. !, :

n

1. Bn = bk ;

k=1

2.a1, a2, ..., an, ... limn®¥an = 0.

" anbn

. .* ( + * )#* anBn, n = 1, 2, ..., 3 * ) #0+ ( # 3 * ) !" an # & #) !" Bn. .3 0 + # 3 / & # + * ) ( + *" (# * # # *" (# * /( + anbn # (an+1 - an)Bn. .# # # #' 7# #** ( + # *" (# * # + & #0 1 #" /( +.

. * |Bk| £ M, k ³ 1. &, 0 #7 ( # *" (# * # an 0

 

"e > 0

$n0 "n > n0 |an| < e.

# n > n0 '( %

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

n+ p

 

 

n+ p

 

 

 

 

 

 

 

 

 

 

n+ p

 

 

 

 

(ak +1 - ak

) Bk

£

ak +1 - ak

 

 

 

Bk

 

£ M ak +1 - ak

 

.

 

 

 

 

 

 

k =n+1

 

 

k =n+1

 

 

 

 

 

 

 

 

 

 

k =n+1

 

 

* ( + * a1, a2, ..., an, ...

, )#* ak+1 - ak, k ³ 1, / 0 * #0 # k # 1

 

n+ p

 

 

 

n+ p

 

 

 

 

 

 

 

 

ak +1 - ak

 

=

(ak +1 - ak )

 

=

 

 

 

k =n+1

 

 

 

k =n+1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

82

com.neevia.http://www version trial Converter Personal Neevia by Created

%#/, 9 4. .#0 /3 #*(#*" &+)#* /(

|an+2 - an+1 + an+3 - an+2 + ... + an+p+1 - an+p| = |an+p+1 - an+1| £

|an+p+1| + |an+1| < e + e = 2e.

#*. 1. * %#/ #0 3 / *" (# * # )#* + & /(.

(# // +* % # + *, 3 2( */, )

n+ p

"e > 0 $n0 "n > n0 "p ³1 (ak +1 - ak ) Bk < 2M e,

k =n+1

) 1 +#+ # # 7# *" (# * # /( (an+1 - an)Bn, 0 )#, # /( anbn. 2 ( 0 .

4. ( *

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

3. !, :

1.bn ;

2.a1, a2, ..., an, ... .

83

com.neevia.http://www version trial Converter Personal Neevia by Created

%#/, 9 4. .#0 /3 #*(#*" &+)#* /(

" anbn .

. .* |ak| £ M, k ³ 1. &, 0 #7 # #' 7# *" (# * # /( bn

 

 

 

 

 

n+ p

 

"e > 0

$n0 "n > n0 "p ³1

bk

< e.

 

 

 

 

 

k =n+1

 

n+ p

 

 

 

 

 

3 0 )# Bpn = bk . 1 #" 3 0 ) #/" * (

+ * + # +#(

k =n+1

 

 

 

 

 

 

 

Bn

 

< e, n > n , p ³1.

 

 

 

 

 

p

 

0

 

 

n+ p

. # # 3 0 + # 3 / *

k =n+1

ak bk # )#

n+ p

n+ p -1

(ak +1 - ak ) Bkn .

ak bk = an+ p Bpn -

k =n+1 k =n+1

# n > n0 '( %

 

 

n+ p

 

 

 

 

 

 

 

 

n+ p -1

(ak +1 - ak ) Bkn

 

 

 

 

 

ak bk

=

an+ p Bpn -

 

£

 

 

 

 

k =n+1

 

 

 

 

 

 

 

 

k =n+1

 

 

 

 

 

 

 

 

n+ p-1

 

 

 

n+ p -1

an+ p

 

Bpn

 

+

 

ak +1 - ak

 

 

 

Bkn

 

< M e + e ak +1 - ak

 

.

 

 

 

 

 

 

 

 

 

 

 

k =n+1

 

 

 

 

 

 

 

 

 

k =n+1

 

 

* ( + * a1, a2, ..., an, ...

 

, )#* ak+1 - ak, k ³ 1, / 0 * #0 # k # 1

n+ p -1

 

 

 

 

n+ p -1

(ak +1 - ak )

 

 

 

 

 

 

 

 

ak +1 - ak

 

=

 

=

 

 

k =n+1

 

 

 

 

k =n+1

 

 

84

com.neevia.http://www version trial Converter Personal Neevia by Created

%#/, 9 4. .#0 /3 #*(#*" &+)#* /(

|an+2 - an+1 + an+3 - an+2 + ... + an+p - an+p-1| = |an+p - an+1| £

|an+p| + |an+1| £ M + M = 2M.

(# // +* % # + *, 3 2( */, )

 

 

n+ p

 

"e > 0

$n0 "n > n0 "p ³1

ak bk

< M e + e 2M = 3M e,

 

 

k =n+1

 

 

 

 

 

) 1 +#+ # # 7# *" (# * # /( anbn. 3 ( 0 .

* + // (+ #+ ( !" #0 , 0 ), ) + * +# + #0 3 / * *# #+ * + * + 6 * +# #0 # #", 0 + * +# #0 3 / + ' 2 * # * 3 * + # * ) ! * +# #0 # #".

 

 

 

n

 

1. bn ;

1. Bn = bk

;

 

k =1

 

 

2. a1, a2, ..., an, ... limn®¥an

2. a1, a2, ..., an, ...

= 0.

.

85

com.neevia.http://www version trial Converter Personal Neevia by Created

%#/, 9 5. .#0 '3, #% #*(#*" &+)#* /(

5. ( #

( 6 / , #0+ * / #0 , '3 #% *" (# * # )#* +!" /( +, /+ / */ * ! ) * ! * ) #0 # #" # * 7 7 & **#) * #" 0 ( ) ## /( +.

4. ! a1 ³ a2 ³ ... ³ an ³ ... ³ 0 limn®¥an = 0. "

(-1)n-1an = a1 - a2 + a3 - a4 + ... + (-1)n - 1an + ...

.

 

 

 

. 3 0 )# b = (-1)n-1, n ³ 1. &(

 

 

 

n

 

 

 

n

n-1

0,

n

Bn = bk =1-1+1 -1+K + (-1)

 

=

n .

k =1

 

1,

( + , * ( + * )#* Bn & #) . # 3 0, * ( + * )#* an # bn ( + + / * +#/ 1 # 2 #0 # #" # 1 /(

(-1)n-1an = anbn

*" (# */. 4 ( 0

.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

. ** # )#* + ' /(

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(-1)n-1

=1-

1

+

1

-

1

+K+

(-1)n-1

+K.

 

 

 

 

 

 

 

 

 

n

2 3

 

4

 

 

 

n

)#* 1/n

* / */ 0,

7 /( *" (# */ #0 , '3 #%. ( & ' * !, /(

 

 

 

 

(-1)n-1

 

 

1

 

 

 

 

 

 

 

 

=

 

 

 

 

 

 

 

 

n

 

 

 

 

 

 

 

 

n

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(* + / * 3 ' & #) * #' /(, !' *" (# */.

86

com.neevia.http://www version trial Converter Personal Neevia by Created

,%#/ 9 5. .#0 ,'3 #% *"(#*# )#*+& /(

( + , /( + ** # *" (# */, *" (# */ 3* . # 3 0, ! * # # * '

# * + *" (/6 & */ /(, ) + #)#+ # * ## * + *" (/6#"*/ /( +, / 0!+ */

+! 2( ' # + * * ( 2 '.

87

com.neevia.http://www version trial Converter Personal Neevia by Created

 

 

, %#/ 10

 

 

1. ) *+ '* + *" (/6 & */ )#* + & /(

 

10

 

1. ) *+ '* + *" (/6 & */ )#* + & /( ......................................................................................................

88

2.

.* # *+ '* + 3* *" (/6 & */ )#* + & /( ............................................................................

89

3. # ......................................................................................................................................................................

91

4. .

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

94

5. .

/ # 3 * ) #0+ ( ## ................................................................................................................................

95

1." &

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

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

** + # * 3

(a1 + a2 +K+ an

) + (an +1

+ an +2

+K+ an

)

+K + (an

+1

+ an

+2

+K+ an ) +K.

1

1

1

2

 

k -1

 

k -1

 

k

( 6 / * + #+ + 0 2

*, /

 

0!+ */ * ) ! *+ '* + *" (/6 & */ /(.

1. ) an ,

 

 

 

 

 

 

 

(1)

 

(ank -1 +1 + ank -1 +2 +K+ ank )

 

 

 

.

. " (# * /( an * S 0 ), ) * ( + * S1, S2, ..., Sn... & ) * !" * # S *+ # (. .* ( + * ) * !" * /( (1) # +#(

Sn1 , Sn2 ,K, Snk ,K .

* /+ / */ ( * ( + * * ( + * # S1, S2, ..., Sn, ... . ( * ( + * *" (/6 '*/

* ( + * # *" (# */ 2 (. .1 /( (1) *" (# */ * S, ) ( 0!+ 1.

88

com.neevia.http://www version trial Converter Personal Neevia by Created

, %#/ 10 2. .* # *+ '* + 3* *" (/6 & */ )#* + & /(

2. ( " - &

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

(2)

ank = an1

+ an2

+K+ ank +K,

 

&( n1, n2, ..., nk, ... - * ( +

* !" )#*, + ' 2( )#* +* ) */ + (#

0. ( '* +# * 6 * +#

( / +*/ & *" (/6 & */ /(, #7 ( / 3* *" (/6 & */. ( 6 /

+! 2 * # *+ '* + 3* *" (/6 & */ /(.

 

 

2. ) an , ank

.

 

. .* * ) an ³ 0 ( / +* " n ³ 1 # /( an *" (# */ S. &( * ( + * & ) *

!" * Sn

3!+ #

 

 

 

 

 

 

 

 

 

 

lim Sn

= sup Sn = S.

 

 

 

 

n®¥

n³1

 

 

 

m

 

 

 

 

 

 

 

3 0 )# ) 0 sm = an ) * * /( (2) #

 

 

 

 

k

 

 

 

 

 

 

 

k =1

 

 

 

 

 

 

 

 

N = max {an , an

,K, an } .

 

 

 

 

 

1

2

m

 

. # 1 #" 3 0 ) #/" * + ( #+ + * +

 

 

 

 

 

 

 

 

 

sm £ SN £ S,

 

 

* ) * / * SN * ( 2# +* * & !, + 7 (7# + ) * * sm. ( + ,

/

* ( + * ) * !" * sm & #) # 1 /( (2) *" (# */, #)

 

 

¥

 

 

 

 

 

 

 

an

 

= S ¢ = lim sm = supsm £ S.

 

 

k =1

k

m®¥

m³1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

89