# Math Help - Power series

1. ## Power series

Does anyone know how to find the power series of a^x ?
and/or
Show that e^s+t = e^se^t using algebra with power series?

2. Given two series $\sum\limits^{\infty}_{k=0}a_{k}$ and $\sum\limits^{\infty}_{k=0}b_{k}$, the Cauchy product is

$\sum^{\infty}_{k=0}c_{k}$
where

$c_{n}= \sum\limits^{n}_{k=0}a_{k}b_{n-k}\;\;n \in N.$

we have
$e^x=\sum^{\infty}_{k=0}\frac{x^{k}}{k!}$
and
$e^y=\sum^{\infty}_{k=0}\frac{y^{k}}{k!}$
and by the cauchy product of $e^x .e^y$

$c_{n}=\sum^{n}_{k=0}\frac{x^{k}y^{n-k}}{k!(n-k)!}=\frac{(x+y)^{n}}{n!}$
so
$e^{x+y}=e^{x}.e^{y}.$

3. Wow thank you so much, we haven't touched on the Cauchy Product yet in class but it makes this problem much easier.