Show that if $|q| < 1$, then $\displaystyle{\sum_{n=1}^{\infty}} \frac{\mu(n)*q^n}{1-q^n} = q$.
I have a feeling that $\displaystyle{\sum_{n=0}^{\infty}}q^n = \frac{1}{1-q}$ (for $|q|<1$) is going to make an appearance, but from there I am not sure.
As $n$ gets larger, the denominator will approach $1$ and the nominator will approach $0$. But I am not sure if that helps, since this is a sum, not a limit. I can't see how that would help me get $q$.
The answer is pretty simple: $$\sum_{n=1}^{\infty} \frac{\mu(n)q^n}{1-q^n} = \sum_{n=1}^{\infty}\sum_{t=0}^{\infty}\mu(n)q^{nt} = \sum_{n=1}^{\infty}\sum_{d|n}\mu(d)q^{n}=q$$ Caused by the simple lemma:
For $n>1$ the $$\sum_{d|n}\mu(d)=0$$ and for $n=1$ $$\sum_{d|1}\mu(d)=1$$
In general this is the particular case of Lambert series.
https://en.wikipedia.org/wiki/Lambert_series