Example of series where $\sum a_n$ convergent but $\sum n {a_n}^2$ divergent?

Question

Can anyone suggest an example of a sequence $\{a_n\}_n\subset (0,\infty)$ such that $\sum a_n$ is convergent but $\sum n {a_n}^2$ is divergent?

Answer 1

Since we must have $a_n = \Omega(\frac{1}{n})$ we think about sparsness, so something like: $a_{2^m}=\frac{1}{m^2+1}$, all other a's being very small, say $a_n = 2^{-n}$ if $n$ is not a power of two, will do since then for $n=2^m$, $na_n^2 = \frac{2^m}{(m^2+1)^2}$ which obviously goes to infinty, so the series $\sum n {a_n}^2$ is divergent