Given the series $\displaystyle\sum^\infty _{n=1}\frac{a_n}{n}$, where $a_n>0$, converges. Prove that $$\sum^\infty_{k=1}{\sum ^\infty _{n=1}\frac{a_n}{n^2+k^2}}$$ converges.
My idea is $$\sum^\infty_{k=1}{\sum ^\infty _{n=1}\frac{a_n}{n^2+k^2}}=\sum^\infty_{n=1}{a_n\sum ^\infty _{k=1}\frac{1}{n^2+k^2}}.$$
I have tried to make $$\sum ^\infty _{k=1}\frac{1}{n^2+k^2} \leq \frac{\alpha}{n}$$ but I did not make it.
Help me, thank you so much!