Proving that $\sum_{n=1}^{\infty} \left(\frac{a_1^{1/s}+a_2^{1/s}+\cdots +a_n^{1/s}}{n}\right)^s$ converges when $\sum_{n=1}^{\infty}a_n $ converges

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.

Solution 1:

It converges by Hardy's inequality: $$\sum_{n=1}^{\infty} \left(\frac{a_1^{1/s}+a_2^{1/s}+\cdots +a_n^{1/s}}{n}\right)^s\leq \left(\frac{s}{s-1}\right)^s\sum_{n=1}^{\infty} a_n.$$