Power Series Expansion for Hyperbolic Sine Function

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Theorem

The hyperbolic sine function has the power series expansion:

\(\ds \sinh x\) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \frac {x^{2 n + 1} } {\paren {2 n + 1}!}\)
\(\ds \) \(=\) \(\ds x + \frac {x^3} {3!} + \frac {x^5} {5!} + \frac {x^7} {7!} + \cdots\)

valid for all $x \in \R$.


Proof

From Derivative of Hyperbolic Sine:

$\dfrac \d {\d x} \sinh x = \cosh x$

From Derivative of Hyperbolic Cosine:

$\dfrac \d {\d x} \cosh x = \sinh x$


Hence:

\(\ds \dfrac {\d^2} {\d x^2} \sinh x\) \(=\) \(\ds \sinh x\)


and so for all $m \in \N$:

\(\ds m = 2 k: \ \ \) \(\ds \dfrac {\d^m} {\d x^m} \sinh x\) \(=\) \(\ds \sinh x\)
\(\ds m = 2 k + 1: \ \ \) \(\ds \dfrac {\d^m} {\d x^m} \sinh x\) \(=\) \(\ds \cosh x\)

where $k \in \Z$.


This leads to the Maclaurin series expansion:

\(\ds \sinh x\) \(=\) \(\ds \sum_{r \mathop = 0}^\infty \paren {\frac {x^{2 k} } {\paren {2 k}!} \map \sinh 0 + \frac {x^{2 k + 1} } {\paren {2 k + 1}!} \map \cosh 0}\)
\(\ds \) \(=\) \(\ds \sum_{r \mathop = 0}^\infty \frac {x^{2 k + 1} } {\paren {2 k + 1}!}\) $\map \sinh 0 = 0$, $\map \cosh 0 = 1$

From Series of Power over Factorial Converges, it follows that this series is convergent for all $x$.

$\blacksquare$


Also see


Sources

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