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 $\lim (na_n)=0$ ?
Asked
Active
Viewed 87 times
0
-
See this: http://math.stackexchange.com/questions/55649/does-f-monotone-and-f-in-l-1a-infty-imply-lim-t-to-infty-t-ft-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 Answers
1
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 $n > N$
RRL
- 90,707