Show: if $\sum_{n>0} f(n)$ is convergent, then $\sum_{n>0} n^{1/n}f(n)$ is convergent

Question

If $\displaystyle{\sum_{n>0} f(n)}$ is convergent, then show that $\displaystyle{\sum_{n>0} n^{1/n}f(n)}$ is convergent .

I am trying using Abel's test, but I can't find my way.

Answer 1

Show that $\frac 1 x (\log\, x)$ is decreasing on $(e,\infty)$ by showing that the derivative is negative. Conclude that $n^{1/n}$ is decreasing for $n >e$. Also $n^{1/n} \to 1 $. Now apply Abel's test ignoring the first two terms of the series.