I'm trying to show that the complete elliptic integral of the second kind can be represented as: $$E(k)=\frac{\pi}2 \left(1-\sum_{n=0}^{\infty}\left(\frac{(2n-1)!!}{(2n)!!}\right)^2\frac{k^{2n}}{2n-1}\right)$$
I first used the binomial theorem to get: $$E(k)=\int_0^\frac{\pi}2\sum_{n=0}^\infty\binom{\frac{-1}2}nk^{2n}\sin^{2n}\theta d\theta$$
I then used Wallis' identity to evaluate the integral: $$\sum_{n=0}^{\infty}\binom{\frac{-1}2}nk^{2n}\left(\frac{\pi}2\right)\frac{(2n-1)!!}{(2n)!!}$$
I'm a mechanical engineer, so I'm not well-versed in real analysis. What are some tricks I could use to combine my binomial and my double factorials? Should I express them both as infinite products and combine them? All input is appreciated. Thanks!
EDIT: Let me clarify the question I'm asking. I'm really just trying to show that the complete elliptic integral of the second kind can be shown as the series above; I'm not trying to demonstrate any relationships between it and any of the other elliptic integrals/functions.
Now, I've come pretty close to determining the derivation of the series, but I've made some mistakes along the way. I will be showing each of the steps I've taken, starting with the integral form and ending with the series.
Beginning with: $$ E(k)=\int_0^{\frac{\pi}2} \sqrt{1-k^2\sin^2\theta}d\theta $$ One can convert the integrand into a binomial series: $$ E(k)=\int_0^{\frac{\pi}2} \sum_{n=0}^{\infty} \binom{\frac12}n(-1)^nk^{2n}\sin^{2n}\theta d\theta $$ Using a form of Wallis' identity, $sin^{2n}\theta d\theta$ becomes $\frac{\pi}2 \frac{(2n-1)!!}{(2n)!!}$, and therefore: $$ E(k)=\frac{\pi}2 \sum_{n=0}^{\infty} \binom{\frac12}n(-1)^nk^{2n} \frac{(2n-1)!!}{(2n)!!} $$ To eliminate the binomial coefficient, $\binom{\frac12}n$ may be (as Lucian pointed out) represented as: $$ \binom{\frac12}n=\frac{(\frac12)(\frac12-1)(\frac12-2)...(\frac12-n+1)}{n!} $$ By multiplying the numerator and denominator by $2^n(\frac12-n)$, one gets $$ \binom{\frac12}n=\frac{(2n-1)!!}{(2n)!!}\frac{(-1)^n}{\frac12-n} $$ By multiplying the numerator and denominator by -2, one gets $$ \binom{\frac12}n=\frac{(2n-1)!!}{(2n)!!}\frac{(-1)^{n+1}2}{2n-1} $$ Replacing the binomial with the above in the series, one gets: $$ E(k)=\frac{\pi}2 \sum_{n=0}^{\infty} (-1)^{2n+1}\left(\frac{(2n-1)!!}{(2n)!!}\right)^2\frac{2k^{2n}}{2n-1} $$ Assuming $n$ is an integer, the exponent of -1 is always an odd parity number, and so the summand is consistently negative. Therefore, the series may be expressed as: $$ E(k)=-\frac{\pi}2 \sum_{n=0}^{\infty} \left(\frac{(2n-1)!!}{(2n)!!}\right)^2 \frac{2k^{2n}}{2n-1} $$ Evaluating the summand at $n=1$ yields -2, so: $$ E(k)=\frac{\pi}2\left(2-\sum_{n=0}^{\infty} \left(\frac{(2n-1)!!}{(2n)!!}\right)^2 \frac{2k^{2n}}{2n-1}\right) $$ Needless to say, those pesky 2s are a bit of a problem. Can you see where I went wrong?
Hint: $\displaystyle{m\choose n}=\dfrac{m(m-1)\cdots(m-n+1)}{1\cdot2\cdot3\cdots n}$ for all $m\in\mathbb C$ and $n\in\mathbb N.~$ Now let $m=-\dfrac12$ and simplify the expression $($by amplifying both numerator and denominator with $2^n$, and then regrouping the terms$)$.