If $||A^N|| < 1$, is $I-A$ invertible?

Question

i have been given this question in functional analysis saying:

Let $X$ be a banach space and $A$ is a bounded linear operator on $X$, and there exists some natural $N$ for which $ ||A^N|| < 1 $. Is $I-A$ invertible?

We need to prove or give a counterexample.

i know it is true when $N=1$ as we proved it in class but for general $N$ i don't know. help appreciated

Answer 1

The comment by Geoff Robinson was really an answer:

You have enough knowledge from class to know that $I-A^{N}$ is invertible, and $$I - A^{N}= (I-A) ( I + A + A^{2} + \ldots + A^{N-2} + A^{N-1})$$