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?
Given two series $\displaystyle \sum\limits^{\infty}_{k=0}a_{k}$ and $\displaystyle \sum\limits^{\infty}_{k=0}b_{k}$, the Cauchy product is
$\displaystyle \sum^{\infty}_{k=0}c_{k}$
where
$\displaystyle c_{n}= \sum\limits^{n}_{k=0}a_{k}b_{n-k}\;\;n \in N.$
we have
$\displaystyle e^x=\sum^{\infty}_{k=0}\frac{x^{k}}{k!}$
and
$\displaystyle e^y=\sum^{\infty}_{k=0}\frac{y^{k}}{k!}$
and by the cauchy product of $\displaystyle e^x .e^y$
$\displaystyle c_{n}=\sum^{n}_{k=0}\frac{x^{k}y^{n-k}}{k!(n-k)!}=\frac{(x+y)^{n}}{n!}$
so
$\displaystyle e^{x+y}=e^{x}.e^{y}.$