Let $\sum\limits_{n=1}^{\infty} a_n$ a series of positive terms convergent. Show that $\sum\limits_{n=1}^{\infty} \frac{a_1+2a_2+...+na_n}{n(n+1)}$ converges to the same value of $\sum\limits_{i=1}^n a_n$
I think I can use the Cauchy condensation test but I would like to know if there is an easier way.