real analysis - How to prove that if $sum|a_j|

I would like to prove the following statement about series.

Suppose that $\{a_j:j\ge1\}$ is a real sequence and $a_j\ne0$ for each $j\ge1$. If the series $\sum_{j=1}^\infty|a_j|<\infty$, then the series

$$\sum_{j=1}^\infty|a_j|\log(|a_j|^{-1})<\infty.$$

It is not difficult to show that this statement is true for some particular sequences $\{a_j:j\ge1\}$. For example, suppose that $a_j=j^{-2}$ and use the integral test for convergence. But I have no idea how to prove the statement for a general sequence $\{a_j:j\ge1\}$.

Any help would be much appreciated!

Answer

Try $a_j=1/(j(\log j)^c)$ for every $j\geqslant3$, then $\log(1/a_j)\sim\log j$ hence $\sum\limits_ja_j$ converges exactly when $c\gt1$ and $\sum\limits_ja_j\log(1/a_j)$ converges exactly when $c\gt2$.

Thus, any sequence defined by $a_j=1/(j(\log j)^c)$ for every $j\geqslant3$, for some $1\lt c\leqslant2$, disproves your claim.