Let $\sum a_n$ be a convergent series of positive terms ; then we know $\lim \inf (na_n)=0$ ; can we derive from here that if $\{a_n\}$ is decreasing , then $\lim (na_n)=0$ ?
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$\begingroup$ See this: math.stackexchange.com/questions/55649/… $\endgroup$– Prahlad VaidyanathanAug 18, 2015 at 16:16
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$\begingroup$ What about $a_n=\frac1n$? EDIT: Oh, you're only talking about sequences for which the sum converges. $\endgroup$– Akiva WeinbergerAug 18, 2015 at 17:41
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1 Answer
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Hint:
For any $\epsilon > 0$ there exists $N \in \mathbb{N}$ such that $\displaystyle\epsilon > \sum_{k=n+1}^{2n}a_k> na_{2n}$ for all $n > N$
