I've been reading generationgfunctionology by Herbert S. Wilf (you can find a copy of the second edition on the author's page here).
On page 33 he does something I find weird. He wants to shuffle the index forward and does so like this: \begin{align*} (f_{n+1})_{n\in N_0} &= \frac{(f(X)-f(0))}{X}\\ (f_{n})_{n\in N_0} &= \sum_{n\in N_0} f_nx^n \end{align*}
Why is this allowed? Namely $X$ has a noninvertible (0) constant term, so how is this division (multiplication with the reciprocal) defined within the ring of formal power series?