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標題: Series of functions & uniform convergence [打印本頁]

作者: kingwinner 時間: 10-4-8 05:15 PM 標題: Series of functions & uniform convergence

Part a 我計到既答案:
S(x) = x^2 / [1- 1/(1+x^2)] = 1 + x^2 if x=/=0
S(x) = 0 if x=0

有冇人可以教下我點做 part b?

Definition: a seqeucne of functions fn converges uniformly to f on S if for all e>0, there exist N such that for all x E S, if n>N, then |fn(x)-f(x)|<e.
Equivalently, sup{|fn(x)-f(x)|: x E S} ->0 as n->infinity.
[相關: Weierstrass M-test, uniform convergence of SERIES of functions]

請高手賜教! thx!

[ 本帖最後由 kingwinner 於 10-4-8 05:16 PM 編輯 ]

作者: Gnoehc 時間: 10-4-9 12:21 AM

Basically the so-called definition is an iff statement.

Rough idea.
Suppose f is a pointwise limit of f_n, f_n converges to f uniformly if
    $\forall \epsilon>0,\exists N\in\mathbb{N},n\ge N ,\forall x\in S\implies |f-f_n(x)|<\epsilon$
Negate the statement above, we have f does not converge uniformly on S if
    $\exists \epsilon>0,\forall n\in\mathbb{N} ,\exists x\in S\implies |f-f_n(x)|\ge \epsilon$

Define
    $S_n =\sum_{k=1}^nf_k=x^2\sum_{k=1}^n\frac{1}{(1+x^2)^k}=1-\frac{1}{(1+x^2)^n}$
then
    $|S_n-1|=\frac{1}{(1+x^2)^n}$

An natural question, does it converge uniformly on R? The answer is no.
Since for any positive integer n (no matter how big), I can take (fixed epsilon < 1)
    $x=\\pm\\sqrt{\\left(\\frac{1}{\\epsilon}\\right)^{1/n}-1}$
such that
    $|S_n-1|=\epsilon$

To avoid the problem (to ensure uniform convergence), we have to avoid having such x. Since uniform convergence cannot take place if x can take an arbitrarily small quantity, then the range |x|>p, p>0 will do!

The sum S(x) will converge to 1 uniformly on any closed inteval [a, b], a>0 or [d, c], c < 0.
Proof.
    $\\sum\\frac{x^2}{(1+x^2)^n}\\leq \\sum\\frac{b^2}{(1+a^2)^n}$
Since right hand side converges, left hand side converges uniformly.

[ 本帖最後由 Gnoehc 於 10-4-9 12:44 AM 編輯 ]

作者: Gnoehc 時間: 10-4-9 12:50 AM

A last remark, the summation of the package mathptmx is really ugly....

作者: kingwinner 時間: 10-4-9 04:04 AM

1) 點計到 Sn = 1 - 1/(1+x^2)^n ?
仲有點解要計 |Sn -1|? 請問個 1 o係邊度黎?

2) ∑ b^2 / (1+a^2)^n converges, is this because it is a "geometric series"?

thx a lot!

[ 本帖最後由 kingwinner 於 10-4-9 07:11 PM 編輯 ]

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