# Power Series Expansion for Error Function

## Theorem

$\displaystyle \map \erf x = \frac 2 {\sqrt \pi} \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {n! \paren {2 n + 1} }$

where:

$\erf$ denotes the error function
$x$ is a real number.

## Proof

 $\ds \map \erf x$ $=$ $\ds \frac 2 {\sqrt \pi} \int_0^x e^{-u^2} \rd u$ Definition of Error Function $\ds$ $=$ $\ds \frac 2 {\sqrt \pi} \int_0^x \paren {\sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {u^{2 n} } {n!} } \rd u$ Definition of Real Exponential Function $\ds$ $=$ $\ds \frac 2 {\sqrt \pi} \sum_{n \mathop = 0}^\infty \frac {\paren {-1}^n} {n!} \paren {\int_0^x u^{2 n} \rd u}$ Power Series is Termwise Integrable within Radius of Convergence $\ds$ $=$ $\ds \frac 2 {\sqrt \pi} \sum_{n \mathop= 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {n! \paren {2 n + 1} }$ Primitive of Power

$\blacksquare$