I'm having trouble succeeding in doing this exercise, where $d(n)$ is the number of divisors of $n$. My first instinct was to use the following line of reasoning: for all $|z|<1$, let:
$$f(z)=\sum_{n=1}^\infty d(n)z^n=\sum_{n=1}^\infty\frac{z^n}{1-z^n}$$
And so simply use the relation $d(n)=\frac{f^{(n)}(0)}{n!}$ or even the Cauchy integral formula, for $r\in]0,1[$:
$$d(n)=\frac{1}{r^n}\int_0^{2\pi}f(re^{i\theta})e^{-in\theta}\ d\theta$$
Yet I still find myself stuck. Is this the right strategy in the first place?
EDIT: I'd like to see a manner of solving this problem that uses my initial ideas, I really believe that they can be exploited since it's an exercise from a particular set wherein this line of reasoning is usually the most common strategy. The solution presented in the other post, while completely correct and elegant in its own right, isn't really what I'd expect to see as a solution.
