Let $\left\lbrace x_{n} \right\rbrace _{n \geq 0}$ be a sequence which converges to $0$. Is the series $\sum\limits_{n \geq 0} \dfrac{x_{n}}{n}$ convergent or not?
In fact the question is for which $p \in \mathbb{R}$, $\sum\limits_{n \geq 0} \dfrac{x_{n}}{n^{p}}$ converges? I can show that when $p>1$ then it converges since $\sum\limits_{n \geq 0} \dfrac{1}{n^{p}} < + \infty$ and for $p<1$ I have some counter-examples. But I'm not sure about the case $p=1$.
Just take $x_n = \frac{1}{\log(n)}$ ! By Bertrand's theorem for series :
$$ \sum_{k=2}^\infty \frac{1}{k \log(k)} = \infty $$