Does every power series have an inverse?

Question

I'm not even sure if every polynomial has an inverse, and what restrictions are required on them to have inverses. For example, the people in this question seem to suggest that the polynomial needs to be increasing to have an inverse.

My main question is, does every power series have an inverse?

Answer 1

According to Lagrange Inversion Theorem, the inverse of an analytic function near $x=f(a)$ is given by

$$f^{-1}(x)=a+\sum_{n=1}^\infty\left\lbrace\lim_{w\to a}\left[{(x-f(a))^n\over n!}{d^{n-1}\over dw^{n-1}}\left({w-a\over f(w)-f(a)}\right)^n\right]\right\rbrace$$

Answer 2

This link on the inversion of power series may help you: http://www.ams.org/mcom/1947-02-020/S0025-5718-1947-0022717-X/S0025-5718-1947-0022717-X.pdf

Answer 3

Every polynomial is a power series, just with infinitely many zero coefficients.

If you believe that not every polynomial has an inverse, then you already know that not every power series has an inverse, because the polynomials which lack inverses are already counterexamples.