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文章是讲一致收敛和微分的~我英语不好看不明白~请大家帮帮忙啊~~~~ Theorem 9.13. Assume that each term of is a real-valued function having a finite derivative at each point of an open interval9 s9 K" v( }" L G
. Assume that for at least one point
( w8 s) H2 e2 o) I9 t, S8 B in8 [& \) M9 f6 f- O/ f
the sequence converges. Assume further that there exists a function g such that/ s0 i; m, f( d" S
uniformly on
; c3 L% T- l+ u. I+ ]' ~ . Then:
4 ]1 K" L5 v( o2 K2 w7 }; |$ i
a) There exists a function f such that
B% u! a. {2 q uniformly on9 z5 \2 m& o6 }" X% l. g, I
. b) For each x in
" ~9 q+ c$ A+ t9 d: ?* m the derivative
" O0 a$ W) c6 U exists and equal7 p" W ^( n2 R5 T
. Proof. Assume that
7 c* D* W2 |* h y and define a new sequence/ x+ \9 S6 H) T! ~& _& A+ n
as follows:
! F+ K! y! @. M7 C% M
6 }- @# s' I9 m9 x2 @ (8)
; n0 T% _: ]( @9 K. @! k
The sequence* D8 ?7 Q1 ]& Y$ R d
so formed depends on the choice of c. Convergence of follows from the hypothesis, since
6 `& F/ t: F0 K . We will prove next that
# \( r& W5 v0 d0 [ converges uniformly on$ J- w B+ Z5 I4 @7 m; s" ~
. If , we have + U- A/ E; n( r* f# O6 y
,2 e5 Q3 l, G! Z! l* ?6 z; ?
(9) _4 e; F: m5 ^6 A- k3 x
where' E9 P0 x- U# S4 ?/ d+ v
. Now
& b- @: s5 U! S4 | S, M5 U; C& x exists for each x in5 Z+ b& N5 `* \6 ? W, q5 c: M2 O
and has the value
7 M4 o) F$ l* a$ N' M . Applying the Mean-Value Theorem in (9), we get ,
" @9 l& h, ~' R/ b) r
- o# q' f& [; \0 U (10) where
4 w# G. r3 b4 l9 } lies between x and c. Since
/ k7 B1 ]7 B+ n' } converges uniformly on
( v. u& m* U' j4 r; q (by hypothesis), we can use (10), together with the Cauthy condition, to deduce that* d+ r$ N8 k9 O) J& Z
converges uniformly on
8 q5 s7 w6 T+ w" k, ]' F# `* s . Now we can show that
$ P! G6 f; U& B+ ~7 X converges uniformly on; R# h" ^* @+ I. x% n. ?
. Let us form the particular sequence
1 v- J8 o9 u8 ~$ x5 X corresponding to the special point+ M8 G8 G% t% v0 ?3 n
for which
P z7 U! n) e' h/ r4 s9 s is assumed to converge. Form (8) we can write
an equation which holds for every x in
4 v/ [) q! ^9 F5 f& r" O . Hence we have
This equation, with the help of the Cauthy condition, establishes the uniform convergence of on, H8 C q0 D9 H
. This proves (a). To prove (b), return to the sequence
$ Z% @* M; z! v/ f defined by (8) for an arbitrary point c in
5 q- ]; ^6 e3 Y and let9 N2 o) {$ R7 x. l2 F, P) z
. The hypothesis that' a3 G3 ?# D4 m: M* A2 b6 Z7 q0 w
exists means that . In other words, each/ g7 j c" j, T: H8 m; m: l
is continuous at c. Since- i" {9 m: N4 q! V" Y
uniformly on
8 w' G) ]! H# `) l , the limit function G is also continuous at c. This means that 2 L6 M" _# q0 t; l X" `
(11) the existence of the limit being part of the conclusion. But for" i5 u$ k( x! {: s0 c
, we have
Hence, (11) states that the derivative+ P3 r/ Z' f% C; C! i" n$ @
exists and equals9 J7 J5 [2 O4 p. I' j! y% ^
. But
hence% G4 u7 Y$ A, J: |7 \& H0 J
. Since c is an arbitrary point of
2 F. ]3 G4 M$ A& L8 D , this proves (b). When we reformulate Theorem 9.13 in terms of series, we obtain & W9 C; M# q& R6 V8 V3 l ?; l7 }
Theorem 9.14. Assume that each
8 O' G- |8 m% a2 _ v4 f is a real-valued function defined on! T$ [& k" L0 a
such that the derivative0 J# G8 P( c8 ? n9 ^$ D/ A" X/ ~
exists for each x in
1 u. J$ B' H4 B# o . Assume that, for at least one point/ J6 G* L9 |5 O
in) d' }! @; V1 V0 Y7 [1 G: k1 a1 n
, the series' ] s) v. l6 R2 R/ k
converges. Assume further that there exists a function g such that (uniformly on, y. m9 g( n5 D3 r
). Then: a)! e; ~: [3 c; k A9 y0 P& f
There exists a function f such that
& R7 \8 ~( f5 t# ]% b) k5 C (uniformly on
- J+ O" c% c. Y ). b)6 S' a5 K! J4 R( S( j, L4 X x
If , the derivative* v+ y) A* c0 J2 [' I! j. m
exists and equals
2 \' D3 z; G# D0 Q/ b' w4 A . |