Does $\sum\limits_{n=1}^\infty {{a_n}{\ln{a_n}}}$ converge?

Question

If $\;\sum\limits_{n=1}^\infty {a_n}\;$ converges,

does the following also converge? $$\sum_{n=1}^\infty {a_n}{\ln{a_n}}$$

When I first saw this problem, it was easy.

So I tried comparison test and limit comparison test.

But i couldn't solve it.

Could you help me?

Answer 1

First of all, we have to assume that $a_n>0$ for each $n$, otherwise $\log a_n$ wouldn't make sense. The problem is that $a_n$ can go to $0$ fast enough to make the series $\sum_n a_n$ convergent, but the fact that $\log a_n$ goes to $-\infty$ can spoil the convergence.

Indeed, define $a_n:=\frac 1{n(\log n)^2}$. Then $\sum_n a_n$ is convergent. On the other hand,
$$|a_n\log a_n|\sim \frac 1{n\log n},$$ hence the series $\sum_n a_n\log a_n$ is divergent ($a_n\log a_n\leqslant 0$).