Assume that $a_n\ge0$ such that $\sum_{n=1}^{\infty}a_n $ converges, then
show that for every $s>1$ the following series converges too: $$\sum_{n=1}^{\infty} \left(\frac{a_1^{1/s}+a_2^{1/s}+\cdots +a_n^{1/s}}{n}\right)^s.$$
I failed to handle this with Hölder inequality. Any tips or hint will be appreciated.
Also it might be helpful to see that there is a Césaro sum of $(a_n^{1/s})_n$ appearing in the last series.