Example of a zero sequence $x_n$, such that $\sum_{k=1}^\infty \frac{x_n}{n}$ diverges.

Question

As stated in the title:
Let $x_n$ be a zero sequence, such that $\sum_{k=1}^\infty \frac{x_n}{n}$ diverges.

• either: Find a sequence $x_n$ for which the condition is met.
• or: Prove that such a sequence cannot exist.

So far I have tried several known zero sequences: $\frac{1}{n},(\sqrt[n]{n}-1),\frac{n!}{n^n}$ but none seemed to work.
I also did not manage to prove that the statement is wrong. (I only managed to prove that for every zero sequence $x_n$ the series $\sum_{k=1}^\infty \frac{x_n}{n^2}$ converges.)

I don't think blindly trying several zero sequences is the best idea, so I would like to know how to approach this problem.

Answer 1

What about $$x_n = \frac{1}{\ln(n+1)} \quad ?$$