Let $(x_n)_{n\in\mathbb N}$ be a non-negative, non-increasing sequence such that $\sum_{n=1}^{\infty} x_n<\infty$. I want to show that $$\lim_{n\to\infty}\{nx_n\}=0.$$ I’ve definitely seen this question asked on this forum before, but I’m unable to find it now. Any reference or hint would be appreciated.
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Hint. Can you check that $2(a_{\lfloor n/2 \rfloor} + \cdots + a_n) \geq n a_n$? – Sangchul Lee Apr 08 '16 at 21:31
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1Answer here: http://math.stackexchange.com/a/1719134/148510 – RRL Apr 08 '16 at 21:31
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Another interesting question is if $o(1/ (n \log n))$ necessary.http://math.stackexchange.com/a/1719134/148510 – RRL Apr 08 '16 at 21:37
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http://math.stackexchange.com/q/1697300/148510 – RRL Apr 08 '16 at 21:39
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@RRL Thank you, that’s precisely the thread I’ve been looking for. – triple_sec Apr 08 '16 at 21:42
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You're welcome. The second link has a very nice answer by @Etienne – RRL Apr 08 '16 at 21:44
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If $(a_n)$ is a decreasing sequence of strictly positive numbers and if $\sum{a_n}$ is convergent, show that $\lim{na_n}=0$ – robjohn Apr 08 '16 at 21:55