If $\sum_{n=1}^\infty a_n$ converges, prove that $\lim_{n\to \infty} (1/n) \sum_{k=1}^n ka_k = 0$. the series $a_n$ is any arbitrary converging series.
My thought process was that the $1/n$ will definitely go to zero as n approaches infinity; however, the series $k*a_k$ seems to approach infinity at the same rate. This confuses me because if both the numerator and denominator approach infinity, I think that the overall equation will equal 1, not zero.
I also considered that the sum is geometric, but I do not believe that is the case, because both $a_k$ and $k$ are variables that are not constant. I was thinking that I could argue if the series $a_k$ converges to a point, eventually the terms of the sequence will be so close to each other that it resembles a constant multiplier, but that argument is not very strong.
Any tips, hints, or leads at the solution are appreciated!

Let $x_n=\sum_{k=1}^n a_k$. The hypothesis is that $x_n$ converges to some $x$. Note that $\sum_{i=1}^n x_i = \sum_{k=1}^n (n-k+1)a_k$, and recall (or if necessary, prove) that if $y_n\to y$ then $$\hat y_n=\frac{1}{n}\sum_{i=1}^n y_i\to y$$
