Given the following sum:
\begin{align} S=\sum\limits_{n=1}^\infty a_n\log(a_n) \end{align}
And this "contract":
\begin{align} \sum\limits_{n=1}^\infty a_n&=1 \\ a_n&\in(0,1] \end{align}
Does the sum $S$ always converge? I tried a few coefficients (e.g. $a_n=\frac{6}{\pi^2 n^2}$ or $a_n=2^{-n}$), but i only found converging values.
Thank you very much