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

高等微積分^*^

Prove: If Σ(n=1→ ∞) u_n(x) is uniformly convergent for a<=x<=b, then the series is uniformly convergent in each smaller interval contained in the interval a<=x<=b.More generally, if a series is uniformly convergent for a given set E of values of x , then it is uniformly convergent for any set E_1 that is part of E. (過程請詳答)

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  • 1 decade ago
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    由定義即得!

    2009-08-31 22:53:12 補充:

    已知Σ[n=1~∞] u_n(x) = f(x) uniformly in E

    則For all ε>0, there exists N>0 (N is indep. on x), such that

    |Σ[n=N~∞] u_n(x) - f(x) |<ε, for all x in E,

    thus

    |Σ[n=N~∞] u_n(x) - f(x)|<ε, for all x in E_1.

    由紅字部分知 Σ[n=1~∞] u_n(x) = f(x) uniformly in E_1

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