# Power Series Expansion for Sine Integral Function

## Theorem

$\ds \map \Si x = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1} \times \paren {2 n + 1}!}$

where:

$\Si$ denotes the sine integral function
$x$ is a real number.

## Proof

 $\ds \map \Si x$ $=$ $\ds \int_0^x \frac {\sin u} u \rd u$ Definition of Sine Integral Function $\ds$ $=$ $\ds \int_0^x \frac {\paren {-1}^n} u \paren {\sum_{n \mathop = 0}^\infty \frac {u^{2 n + 1} } {\paren {2 n + 1}!} } \rd u$ Power Series Expansion for Sine Function $\ds$ $=$ $\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \paren {\int_0^x \frac {u^{2 n} } {\paren {2 n + 1}!} \rd u}$ Power Series is Termwise Integrable within Radius of Convergence $\ds$ $=$ $\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1} \times \paren {2 n + 1}!}$ Primitive of Power

$\blacksquare$