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

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?

• Did you try $n^{-\alpha}$? – Will M. Mar 12 '19 at 19:45

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