Power Series Expansion for Hyperbolic Sine Function

Theorem
The hyperbolic sine function has the power series expansion:

valid for all $x \in \R$.

Proof
From Derivative of Hyperbolic Sine Function:
 * $\dfrac \d {\d x} \sinh x = \cosh x$

From Derivative of Hyperbolic Cosine Function:
 * $\dfrac \d {\d x} \cosh x = \sinh x$

Hence:

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

where $k \in \Z$.

This leads to the Maclaurin series expansion:

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

Also see

 * Power Series Expansion for Hyperbolic Cosine Function