Prove $\sum\limits_{k=0}^{\infty} x^{k} = \frac{1}{1-x}$ also holds for matrices. I need to prove $\sum\limits_{k=0}^{\infty} x^{k} = \frac{1}{1-x}$ also holds for matrices. Thus the previous should be true when $x$ is a matrix. I honestly have no idea where to start, so any suggestions are welcome.

I'll give hints, which are similar to what the comments suggest. 
To start, we need to suppose that $x$ is square, and $\|x\|<1.$ Call $S_n=I+x+x^2+\cdots+x^n.$ First, we claim that this converges as $n\rightarrow\infty,$ and the series converges absolutely and uniformly. Check why this is true. Call this limit $S$. We can see that $I-x$ and $S_n$ commute. What does their product give us, and what happens in the limit?
