Anonymous
Anonymous asked in 教育與參考考試 · 1 decade ago

非常急!!高等微積分!!

證明that if {fn:R--->R}is a sequence of continuously differentiable functions such that the sequence of derivatives {f'n:R--->R} is uniformly convergent and the sequence {fn(0)} is also convergent,then {fn:R--->R}is pointwise convergent.Is the assumption that the sequence {fn(0)}converges necessary?

請高手相助~!

感激不盡!

Update:

有點看不懂第一個回答=.=...

那是舉例嗎?

還是就證明完成了?不懂不懂~~~~

唉優高等微積分怎麼可以這麼難..........

Update 2:

歐歐歐~!了解了解~!!!太感謝你了!!

Update 3:

前面的証明=.=...

可以幫忙一下嘛?要先證明(1)到(2)是正確的!

感激不盡!

1 Answer

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  • 1 decade ago
    Favorite Answer

    省略{f(n)}收歛, 則不一定成立!

    設 f_n(x)= sin(nx) /n^3 + n

    => f_n'(x)= cos(nx)/n^2 -> f(x) = 0 converges uniformly.

    但 {f_n(x)} diverges.

    2009-05-04 02:14:58 補充:

    題目:(1)f_n(x)連續可微, (2) f_n'(x) 均勻收歛, (3) f_n(0)收歛 => f_n(x)收歛

    是否第(3)個條件可省略?

    我的回答: 舉例說明,第(3)條件不可省略!

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