From Carothers, Chapter 1, Exercise 34:
Suppose that $a_n \geq 0$ and $\sum_{n=1}^\infty a_n< \infty$. Give an example showing that $\lim\sup_{n\to \infty} n a_n > 0$ is possible.
Looking at the sequence $n a_n$, there is a subsequence $n_k a_{n_k}$ that either diverges or converges to some positive number, where $a_{n_k} \to 0$ because the series itself converges. I tend to get stuck at this point; I know that $a_{n_k}$ must decrease slowly relative to $n_k$ but not so slow so that the series doesn’t converge. Does anyone have any tips?
Best Answer
$a_n=\frac 1 n $ when $n =m^{2}$ for some $m$ and $0$ otherwise.