Complete Elliptic Integral of the First Kind as Power Series

Theorem
The complete elliptic integral of the first kind:
 * $\displaystyle K \left({k}\right) \int \limits_0^{\pi / 2} \frac {\mathrm d \phi} {\sqrt{1 - k^2 \sin^2 \phi} } = \int \limits_0^1 \frac {\mathrm d v} {\sqrt{\left({1 - v^2}\right) \left({1 - k^2 v^2}\right)} }$

can be expressed as the power series: