Complete Elliptic Integral of the Second Kind as Power Series

Theorem
The complete elliptic integral of the second kind:
 * $\displaystyle \map E k = \int_0^{\pi / 2} \sqrt {1 - k^2 \sin^2 \phi} \, \rd \phi = \int_0^1 \dfrac {\sqrt {1 - k^2 v^2} } {\sqrt {1 - v^2}} \, \rd v$

can be expressed as the power series:

Proof
From Reduction Formula for Integral of Power of Sine, $\forall i \in \N$:

Hence: