I need to find the inverse of $1-zb(z)$ with $b(z)=\sum_{n=0}^{\infty}b_nz^n$.
I have tried several approaches where I among other things have tried using the methods in my calculus book but nothing seems to quite work.
What would be the best approach to find this inverse (and what is it)?
(I'm not necessarily asking for the full solution - just the approach and the result so I can deduce the answer myself and know that I got the right answer)
Consider the case where $b_1 = 1$ and $b_j = 0$ for all other $j$. Then your function is $f(z) = 1 - z^2$. Since $f(1) = f(-1) = 0$, this function doesn't have an inverse.
Since the problem of finding the inverse cannot be solved for this specific case, it also can't be solved for the general case. You're going to need to tell us more about the coefficients for us to be of any use.