I need help with this:

Let E a Banach Space and B(E) the Banach space of linear and bounded operators from E to E with the usual norm. Let S $\displaystyle $\in$$ B(E).

If the sum $\displaystyle $\sum\limits_{n = 1}^\infty {\frac{{{S^k}}}{{k!}}}$$ convergs to the operator $\displaystyle $e^S$$, show that $\displaystyle $Se^S$$ = $\displaystyle $e^S S$$. And if T,S $\displaystyle \in $ B(E) conmute then $\displaystyle $e^{S+T}$$= $\displaystyle $e^S$$$\displaystyle $e^T$$. In particular prove that $\displaystyle $e^S$$ is invertible.

Please somebody give me a hand.