User:Tohline/Math/EQ EllipticIntegral01
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<math>~\frac{2K(k)}{\pi}</math> 
<math>~=</math> 
<math>~ 1 + \biggl( \frac{1}{2} \biggr)^2k^2 + \biggl( \frac{1\cdot 3}{2\cdot 4}\biggr)^2 k^4 + \biggl( \frac{1\cdot 3\cdot 5}{2^4\cdot 3}\biggr)^2 k^6 + \biggl( \frac{1\cdot 3\cdot 5 \cdot 7}{2^7 \cdot 3}\biggr)^2 k^8 + \cdots + \biggl[ \frac{(2n1)!!}{2^n n!} \biggr]^2 k^{2n} + \cdots </math> 

Gradshteyn & Ryzhik (1965), §8.113 