$n\lvert a_n\rvert \to 0$, then $\sum_{k=1}^n \frac k n\lvert a_k\rvert \to 0$? I am now working on the converse of Abel's THM and found out one proof of the conditional converse of the theorem. The proof says,
Suppose $$n\lvert a_n\rvert \to 0$$ as $$n \to \infty$$
Then, 
$$\sum_{k=1}^n \frac k n\lvert a_k\rvert \to 0$$ as $$n \to \infty$$
But how does this work? Should I use comparison test? I tried to use root test but I found it hard to apply it given the form of the series. Can you help me? I'd really appreciate your help. 

If 
$$
n\lvert a_n\rvert \to 0,
$$
then
$$
\frac 1n\sum_{k=1}^n k\lvert a_k\rvert \to 0,
$$
This may be seen as an application of the Cesaro theorem.
