A question on convergent series of positive terms

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 $\lim (na_n)=0$ ?

• – Prahlad Vaidyanathan Aug 18 '15 at 16:16
• What about $a_n=\frac1n$? EDIT: Oh, you're only talking about sequences for which the sum converges. – Akiva Weinberger Aug 18 '15 at 17:41

1 Answer

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 $n > N$