Power Series Expansion for Hyperbolic Cosine Function

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Theorem

The hyperbolic cosine function has the power series expansion:

\(\displaystyle \cosh x\) \(=\) \(\displaystyle \sum_{n \mathop = 0}^\infty \frac {x^{2 n} } {\left({2 n}\right)!}\)
\(\displaystyle \) \(=\) \(\displaystyle 1 + \frac {x^2} {2!} + \frac {x^4} {4!} + \frac {x^6} {6!} + \cdots\)

valid for all $x \in \R$.


Proof

From Derivative of Hyperbolic Cosine Function:

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

From Derivative of Hyperbolic Sine Function:

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


Hence:

$\dfrac {\d^2} {\d x^2} \cosh x = \cosh x$

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

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

where $k \in \Z$.


This leads to the Maclaurin series expansion:

\(\displaystyle \cosh x\) \(=\) \(\displaystyle \sum_{r \mathop = 0}^\infty \left({\frac {x^{2 k} } {\left({2 k}\right)!} \cosh \left({0}\right) + \frac {x^{2 k + 1} } {\left({2 k + 1}\right)!} \sinh \left({0}\right)}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \sum_{r \mathop = 0}^\infty \frac {x^{2 k} } {\left({2 k}\right)!}\) $\sinh \left({0}\right) = 0$, $\cosh \left({0}\right) = 1$

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

$\blacksquare$


Also see


Sources