Power Series Expansion for Sine Integral Function

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Theorem

\(\ds \map \Si x\) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1} \times \paren {2 n + 1}!}\)
\(\ds \) \(=\) \(\ds \dfrac x {1 \cdot 1!} - \dfrac {x^3} {3 \cdot 3!} + \dfrac {x^5} {5 \cdot 5!} - \dfrac {x^7} {7 \cdot 7!} + \cdots\)

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

$\blacksquare$


Sources

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