generating functions closed form

Question

I am very new to using generating functions and am trying to apply one here. Is there a nice closed form for the expression $$\sum \frac{x^n}{1-x^n}$$ For example. $\sum x^n=\frac{1}{1-x}$

Answer 1

It is a Lambert series: $$ \sum_{n\geq 1}\frac{x^n}{1-x^n} = \sum_{m\geq 1}d(m)\,x^m$$ where $d(m)$ is the divisor function.

Answer 2

This post elaborates the answer given by Jack D'Aurizio.

Note that $$\frac{x^n}{1-x^n}=\frac{1}{1-x^n}-1=\left( 1+x^n+x^{2n}+x^{3n}+\dotsc \right)-1=x^n+x^{2n}+x^{3n}+\dotsc$$ using the sum for the geometric series (which, according to your post, you already know).

Your function now just sums up these terms over all $n$.

If we want to find out the coefficient of $x^m$ in your function, we have to examine in which of the summands such a term appears. That is, when $n|m$.

Hence, the coefficient of $x^m$ in your function is exactly the number of positive divisors of $m$, also denoted $d(m)$.