Complete Elliptic Integral of the Second Kind as Power Series

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

can be expressed as the power series: