Let $a_n\geq 0 ~(n\geq 1)$ be a non-negative sequence. Suppose $\sum a_n$ converges; can we deduce that $\sum n a_n^2$ converges?
I have no idea how to solve this analysis exercise, can anyone give me some hint? Thanks in advance!
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4Limit comparison test will get you a lot of mileage here – Ninad Munshi Jun 15 '21 at 14:52
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The answer is “yes” if $(a_n)$ is a decreasing sequence: https://math.stackexchange.com/q/2716919. – Martin R Jun 15 '21 at 15:00
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In the general case a counterexample is given in this (deleted) question https://math.stackexchange.com/q/2484070/42969: $a_n=\begin{cases}\frac1{k^2}&\text{if }n=k^4\0&\text{otherwise}\end{cases}$. – Martin R Jun 15 '21 at 15:04
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Oh I understand! Thanks for all answers!! – youknowwho Jun 15 '21 at 15:10