Power Series Expansion for Sine Integral Function

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Theorem

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

where:

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


Proof

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

$\blacksquare$


Sources