# How do you find the Taylor series of f(x)=e^x ?

Sep 21, 2014

Taylor series at $x = 0$ (also called Maclaurin series) for $f \left(x\right)$ is

f(x)=sum_{n=0}^infty{f^{(n)}(0)}/{n!}x^n.

Since if $f \left(x\right) = {e}^{x}$, then

$f \left(x\right) = f ' \left(x\right) = f ' ' \left(x\right) = \cdots = {f}^{\left(n\right)} \left(x\right) = {e}^{x}$,

so,

$f \left(0\right) = f ' \left(0\right) = f ' ' \left(0\right) = \cdots = {f}^{\left(n\right)} \left(0\right) = {e}^{0} = 1$

Hence, the Maclaurin series is

f(x)=sum_{n=0}^infty 1/{n!}x^n=sum_{n=0}^infty x^n/{n!}