Матан Лекции
.PDFcom.neevia.http://www version trial Converter Personal Neevia by Created
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, %#/ 5 |
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4. ." ( ( + |
+ * + |
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* # 2 * + |
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,..., |
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2 +# " ( ( + |
+ * +. |
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2. ! limn®¥xn = l limn®¥yn = p n
xn £ yn.
" l £ p.
. (# ) * # ( #/ ( +, / !" + $# + ! 2,
"e > 0 $n1 "n > n1 |xn - l| < e
#
50
com.neevia.http://www version trial Converter Personal Neevia by Created
, %#/ 5 5. " (# * * ( + * #, & #) ' (+ / * ( + * / # * (# +! # ( #
"e > 0 $n2 "n > n2 |yn - p| < e.
: / +! #/ 3 #" + * +, 3 0 )# n0 = max{n1, n2} # 3 ( ** #+ n > n0. #0 2( * +
l - p = (l - xn) + (xn - yn) + (yn - p),
+ + ' ) * # & + / # / * 3 # |
+ *" (/ e, + / |
+ *" (# /, * (,) |
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. 1 %## 3 ) l - p £ 0, ) ( 0!+ 2.
, 1 %## 3 /+ / */ + * ) * ! * ) ! 2, 1/10n®0 # n®¥.
-#, ) * +# ! 2 ( * * 3 # + +! |
## + * + xn £ yn, ( * ) 3! |
* + ( #+! ( / +* " !" n, )# / * & |
n0. |
5. " ", "
$# 6 ( + * + ", 0 + " * ) /".
3. ! limn®¥xn = limn®¥zn = l n
xn £ yn £ zn.
" limn®¥yn = l.
51
com.neevia.http://www version trial Converter Personal Neevia by Created
, %#/ 5 5. " (# * * ( + * #, & #) ' (+ / * ( + * / # * (# +! # ( #
#*. 1. " (# * * ( + * #, & #) ' (+ / * ( + * / # * (# +! # ( #.
3 #0+ * *+ # # $ ! # 0+ #/ #. , + **#'* ( + ## * !+ (+ " # #%# ", + & /0!) !" ) 3 # " ' ( 0+ # ! *1 (+#).
. (# ) * # ( #/ ( +, / !" + $# + ! 3,
"e > 0 $n1 "n > n1 |xn - l| < e
#
"e > 0 $n2 "n > n2 |zn - l| < e.
: / +! #/ 3 #" + * +, 3 0 )# n0 = max{n1, n2} # 3 ( ** #+ n > n0. #0 1 #" (+ "
+ * + # * 7 #/ xn £ yn £ zn +!+ (# % ) + * +
-e < xn - l £ yn - l £ zn - l < e.
!( // * ( 0+ # (+ ' #", *$# 0 # 6 ( 2 #
"e > 0 $n0 "n > n0 |yn - l| < e,
0 ), ) limn®¥yn = l # ( 0!+ 3.
) (!( 6# 0 #, ) * +# ! 3 ( * * 3 # + +! ## + * + xn £ yn £ zn, ! ( * ) 3! * + ( #+! # ( / +* " !" n, )# / * & n0.
52
com.neevia.http://www version trial Converter Personal Neevia by Created
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, %#/ 6 |
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1. .( * ( + * *" (/6 '*/ * ( + * # |
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6 |
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.( * ( + * *" (/6 '*/ * ( + * #........................................................................................ |
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2. |
* #) ! ( ! * ( + * #.................................................................................................................... |
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3. |
6 * + + # + " & ( & #) ' * ( + |
* #.................................................................... |
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4. |
" (# * * ( + * #, # 6 ' + ! + " #' # |
#2 #' ( ! ................................................. |
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1. ( " " & "
) 2, ) +* ( * ( + * # *" (/6 '*/ * ( + * # + ( * 3/ " 2# 3 0.
1. ) l,
l.
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"e > 0 $n0 "n > n0 |xn - l| < e. |
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** # ( * ( + * x , x ,..., x |
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"e > 0 $k0 |
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nk |
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limk ®¥ |
xn = l, # ( 0!+ 1. |
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1, 0, 1, 0, ..., 1, 0, ... . |
53
com.neevia.http://www version trial Converter Personal Neevia by Created
, %#/ 6 3. 6 * + + # + " & ( & #) ' * ( + * #
8 * ( + * *" (# */. +( + /* + ( 3 3* 2( #, 3 6 +* )# # #' 7#, & + ( + 2( 6#' *" (# * * ( + * # (1). * ( + * (1) # *" (/6# */ ( * ( + * #
1, 1, ..., 1, ... # 0, 0, ..., 0, ... .
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, l.
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* # 2 * + X ) * #) !" ( + * ( + * # & #) , X # + " # #2 & #, !
( (# * %# ! 0+ #/.
( # 2. ' ( x1, x2, ..., xn, ...
limn®¥ xn lim sup xn. ' (
x1, x2, ..., xn, ... limn®¥xn
lim inf xn.
3. & "
2 * 6 * + + #/ + " & ( & #) ' * ( + * #.
2. .
. .* * ( + * x1, x2, ..., xn, ... & #) . 2 * + 1 + # 3 & ( 2 * + # + " & . 3 0 )#
54
com.neevia.http://www version trial Converter Personal Neevia by Created
, %#/ 6 3. 6 * + + # + " & ( & #) ' * ( + * #
a1 = sup xn = sup{ x1 , x2 ,..., xn ,...} ,
n³1
a2 = sup xn = sup{ x2 , x3 ,..., xn ,...} ,
n³2
..........................................................
a |
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= sup x |
= sup |
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a1 ³ a2 ³ ... ³ ak ³ ... . |
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& k. - )#, |
+ 0 * 6 / * ( + * a1, a2, ..., ak, ... |
& #) * #0 # 1 *" (# */. 3 0 )# |
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a = limak |
= inf ak . |
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k ®¥ |
k ³1 |
. 2, ) a = limn®¥ xn . / 1 & 3 (# */, +-+!", ) a /+ / */ ) * #) ! ( * ( + * # x1, x2,
..., xn, ..., +-+ !", ) 3 ' ( & ' ) * #) !' ( 1 ' * ( + * # + *" (# a.
55
com.neevia.http://www version trial Converter Personal Neevia by Created
, %#/ 6 3. 6 * + + # + " & ( & #) ' * ( + * #
) * * #/ ( * ( + * #, *" (/6 '*/ a. ) +! #7 *+ '* + )#* ak, k Î Õ, + " #" & ' * + * + 6#" 2 * +.
1. "n ³ k xn £ ak; |
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2. "e > 0 $nk ³ k x > a |
k |
-e. |
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. 2# e = 1/k, * (# # *+ '* + 1-2 # )# |
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"k ³1 |
$n ³ k a |
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#*. 1. " (# * ( * ( + * # xn a. |
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. 3 %## 5 0 ), ) * ( + * |
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xn |
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*" (# */ a. .* & ##, ) |
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57
com.neevia.http://www version trial Converter Personal Neevia by Created
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com.neevia.http://www version trial Converter Personal Neevia by Created
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