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Question: Suppose $\sum |a_n|<\infty$. Is it true that $\lim_{n\to\infty}n|a_n|=0$?


Suppose not. Then for some $\epsilon>0$ and for every $N\in{\bf N}$, there exists $n\geq N$ such that $$ |a_n|>\frac{\epsilon}{n}. $$ This is far from enough to conclude that $\sum |a_n|=\infty$. I think there might be counterexamples to the statement. Other than this I don't see what could be useful here.

Note that the monotonicity assumption has been dropped from this classical problem:

Series converges implies $\lim{n a_n} = 0$

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  • $\begingroup$ You're right, I was too quick. $\endgroup$ – Arthur Nov 21 '17 at 15:48
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    $\begingroup$ Suppose we take $a_{2^n} = 1/2^n$ for positive integers $n$, and $a_k = 0$ for all other $k$. $\endgroup$ – Bungo Nov 21 '17 at 15:49
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For a counterexample, take the sequence $$a_n=\cases{\frac 1n & if $n$ is a square\\0& otherwise}$$

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