Power Series Expansion for Exponential of x by Sine of x

Theorem
for all $x \in \R$.

Outline of Proof
It is to be established by induction that:
 * $\left.{\dfrac {\rd^n} {\rd x^n} e^x \sin x}\right\rvert_{x \mathop = 0} = 2^{n / 2} \sin \left({n \pi / 4}\right)$

and using this in a Maclaurin series.