example of convergent series, positive sequence, not decreasing

Question

I would like to see an example of sequence $x_n$ such that $\sum_{n=0}^{\infty} x_n$ is convergent where all $x_n \ge 0$, $x_n$ is not decreasing and $$\lim_{n \rightarrow \infty}n x_n \neq 0$$.

(I changed the initial text because I got confused during writing it, and it didn't make sense)

Answer 1

Set $x_{2^{n}}=1/2^{n}$ and $x_{k}=0$ otherwise, then $2^{n}x_{2^{n}}=1$, so $(nx_{n})$ contains a subsequence which does not converge to zero.