Editing Exponential function (section)

===Power series===

''The exponential function is the sum of the [[power series]]''<ref name="Rudin_1987"/><ref name=":0">{{Cite web|last=Weisstein| first=Eric W.|title=Exponential Function|url=https://mathworld.wolfram.com/ExponentialFunction.html|access-date=2020-08-28| website=mathworld.wolfram.com|language=en}}</ref> 
<math display=block>
\begin{align}\exp(x) &= 1+x+\frac{x^2}{2!}+ \frac{x^3}{3!}+\cdots\\
&=\sum_{n=0}^\infty \frac{x^n}{n!},\end{align}</math>
[[Image:Exp series.gif|right|thumb|The exponential function (in blue), and the sum of the first {{math|''n'' + 1}} terms of its power series (in red)]]
where <math>n!</math> is the [[factorial]] of {{mvar|n}} (the product of the {{mvar|n}} first positive integers). This series is [[absolutely convergent]] for every <math>x</math> per the [[ratio test]]. So, the derivative of the sum can be computed by term-by-term differentiation, and this shows that the sum of the series satisfies the above definition. This is a second existence proof, and shows, as a byproduct, that the exponential function is defined for every {{tmath|x}}, and is everywhere the sum of its [[Maclaurin series]].