Let $\sum\limits_{n=1}^\infty a_n$ be a convergent series in which $a_n \ge 0, \forall n=1,2, ...$. Prove that the series $\sum\limits_{n=1}^\infty \frac{1}{n} (a_n + a_{n+1} + ... + a_{2n-1})$ converges as well.