Show that if $\sum \limits_{n=1}^\infty a_n$ converges then $\lim\limits_{n\to\infty}\frac{1}{n}\sum _{k=1}^n ka_k=0$ Show that if $\sum \limits_{n=1}^\infty a_n$ converges then $\lim\limits_{n\to\infty}\frac{1}{n}\sum \limits_{k=1}^n ka_k=0$
since given that  $\sum\limits _{n=1}^\infty a_n$ converges 
then we can say that $\lim _{n\to \infty} a_n=0$
but i don't get any idea to prove this result how to solve this problem


If $\{x_n\}$ converges to $l$, then $$\lim_{n\to\infty} \frac{x_1+x_2+\cdots+x_n}{n} = l $$

Given $\sum_{n=1}^\infty a_n$ converges to $S$, let $x_n$ be the sequence of patial sum, applying above lemma immidiately gives the result.
