I want to be able to approximate the following in closed form $$L=\sqrt{\gamma ^2+4 \pi ^2 \psi ^2} \sum _{j=0}^{\infty } \frac{\left(\frac{(2 j-1)\text{!!} (2 \pi \psi )^j}{(2 j)\text{!!} \left(\gamma ^2+4 \pi ^2 \psi ^2\right)^{j/2}}\right)^2}{1-2 j}$$ but I would also like to accelerate the series since it takes about 50 terms to converge to the desired accuracy.
This derives from $$L=\frac{2 \sqrt{\gamma ^2+4 \pi ^2 \psi ^2} E\left(\frac{2 \psi \pi }{\sqrt{\gamma ^2+4 \pi ^2 \psi ^2}}\right)}{\pi }$$
So the solution to this problem is very similar to finding an approximation for the circumference of an ellipse.
If I can accelerate the summation before hand the approximation won't need to be so precise. Any hints are greatly appreciated!