Take the tour ×  How can you show that the complete elliptic
integral of first kind $ \displaystyle
K(m)=\int_0^\frac{\pi}{2}\frac{\mathrm du}{\sqrt{1-m^2\sin^2
u}}$ that is the same as a series $$K(m)=\frac{\pi}{2}
\left(1+\left(\frac{1}{2}\right)^{2}m^2 +\left(\frac{1\cdot
3}{2\cdot 4}\right)^{2}m^4 +...+ \left(\frac{(2n-1)!!}{2n!!}
\right )^2m^{2n} + ... \right)$$  increases in m?
share|improve this question The expansion of the integral you
wrote has terms for every odd power of $m$ as well. –  Did Dec 8
'12 at 20:56 Im sorry, can you explain again? something wrong on
the expansion? –  JHughes Dec 8 '12 at 21:58 Yes, something was
definitely wrong with the expansion... But it seems you saw the
problem since you made the necessary correction. Note that this
makes the accepted answer, which addresses (incorrectly) the
original version of your question, a little odd. –  Did Dec 8
'12 at 23:42 yeah yeah, thx, i already correct it. –  JHughes
Dec 11 '12 at 1:24 add comment  1 Answer  up vote 1 down vote
acce