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Apostol 數學分析 習題 9.23

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Let an be a decreasing sequence of positive terms. Prove that the series ∑ansinnx converges uniformly on R if, and only if, nan→0 as n→∞

，我一直認為if direction有問題，網路上看到的證明也覺得沒證出來。

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if 那端基本上就是 Abel's test (summation by parts 那招)加上 $$\displaystyle \sum_{k=1}^{n} \sin kx = \frac{\sin \frac{nx}2 \sin \frac{n+1}2x}{\sin \frac x2} = O(n)$$, for $$x\neq 0$$