Is it true that if $\sum x_n<\infty$ for positive $x_n$ then there is some $\alpha>0$ so that $\sum n^\alpha x_n<\infty$?

Question

Is it true that if $\sum x_n<\infty$ for positive $x_n$ then there is some $\alpha>0$ so that $\sum n^\alpha x_n<\infty$?

I'm just curious with the analogy with $p$ series. If $x_n=1/n^p$ then you can always find such an $\alpha$. What about in general?

Answer 1

No. The series where the general term is $x_n=\frac1{n \log^2 n}$ is a counterexample.