Suppose that the series $\sum_{n=1}^\infty a_n$ is conditionally convergent. Prove that the series $\sum_{n=1}^\infty n^2a_n$ is divergent.
How should I start to prove this? I have absolutely no idea how to go about solving this problem. Thanks in advance!
I'd do a proof by contradiction. Firstly, we assume that $\sum{n^2a_n}$ does converge. This implies that $\lim_{n\to\infty}{n^2a_n}=0$. That is, for all $n>N$ for some $N\in\mathbb{N}$, $|a_n|<1/n^2$. But that implies that $\sum{a_n}$ converges absolutely (because of the direct comparison test). Thus, we have reached a contradiction, and the series $\sum{n^2a_n}$ must diverge. In fact, this can be extended to show that the series $\sum{n^pa_n}$ diverges whenever $p<1$.
