If a series converges, does it converge with additional log term multiplied?

Question

If $\sum_{n} |a_n| < \infty$, is it true that $\sum_{n} |a_n\log(a_n)| < \infty$ if $0 \leq a_n \leq 1$?

I am trying to see if $A$ is trace class operator, then $A \log(A)$ is also trace class provided $0 \leq A \leq 1$.

Answer 1

No. Consider:

$$\sum_{n=2}^\infty {1\over n\log^2 n}<\infty$$

However

$$\log\left({1\over n\log^2 n}\right)=-\log n-2\log\log n$$

so that the series in your question is

$$\sum_{n=2}^\infty \left({1\over n\log n}+{2\log\log n\over n\log^2 n}\right)=\infty.$$