A Lambert series is an expression of the form $\sum_{n=1}^{\infty} a_n\frac{x^n}{1-x^n}$. I know it can be expressed as $\sum_{n=1}^{\infty}b_nx^n$, where $b_n=\sum_{d|n}a_d$, if $|x|<1$. We proceed this way:
$\sum_{n=1}^{\infty} a_n\frac{x^n}{1-x^n} = \sum_{n=1}^{\infty}a_n\sum_{k=1}x^{nk}$
Reindexing by $m=nk$, we have $k=\frac{m}{n}$, so $n|m$ and $1\leq m\leq\infty$, since $1\leq n,k\leq\infty$. But now, I don't know how to arrange the sums in order to get what I'm supposed to get. I would appreciate some help. Also, do I need to prove absolute convergence of $\sum_{n=1}^{\infty} a_n\frac{x^n}{1-x^n}$ to do this procedure?
Thank you very much!