Example of sequence $\langle x_n\rangle$ with $x_n>0$ such that series $\sum x_n$ is convergent but $\langle nx_n\rangle$ is NOT a null sequence.

Question

Please provide an example of sequence $\langle x_n\rangle$ of positive terms such that series $\sum x_n$ is convergent but sequence $\langle nx_n\rangle$ is NOT a null sequence.

I try hard but could not find one such. Please help.

Answer 1

Take $x_{n}=1/n$ if $n$ is a perfect square and $x_n=1/n^2$ otherwise. Then it is easy to see that $\sum x_n$ is convergent but $nx_n\not\to 0$ since $nx_n=1$ whenever $n$ is a perfect square.

Answer 2

Start with any sequence $\langle y_n:n\in\Bbb Z^+\rangle$ of positive terms such that $\sum_{n\ge 1}y_n$ converges and replace $y_{n!}$ with $\frac1{n!}$ for each $n\in\Bbb Z^+$.