Let $X$ be a Banach space and $A\in L(X,X)$. Show that $$\sum_{k=0}^\infty \frac{A^k}{k!} $$ converges in $L(X,X)$. Find an upper bound to the norm of sum.
If a series converges absolutely, then it converges. I could say that $$\frac{1}{k!}\|A^k\|\xrightarrow[k\to\infty]{}0 $$ if $A$ was bounded, is it, though? At any rate, that alone is not enough to show convergence. I'm not sure how I should approach this, the lecture notes seemingly don't give any relevant info on how to show such convergence.