Complete Elliptic Integral of the First Kind as Power Series

Theorem
The complete elliptic integral of the first kind:
 * $\ds \map K k = \int_0^{\pi / 2} \frac {\rd \phi} {\sqrt {1 - k^2 \sin^2 \phi} } = \int_0^1 \frac {\rd v} {\sqrt {\paren {1 - v^2} \paren {1 - k^2 v^2} } }$

can be expressed as the power series:

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

Hence: