As mentioned previously the power series for the complete elliptic integral of the 2nd kind doesn't converge very rapidly when the modulus κ=±1. There is also a problem with using double factorials since they grow rapidly and the number of terms is limited to 150 in Mathcad 11 which is when the numbers approach the upper limit on the numbers that can be represented.
For large n the ratio of the (2n-1)/2n approaches 1 and it is easier to use a recurrence relation to compute the new term of the sum using the previous result. The simple program shown below can be used to sum a larger number of terms of the series.
One finds that the sum of the series is nearly identical with the result the Mathcad got using numerical integration.
Even with the large number of terms there is still some error near κ=1.
The error is approximately 1/4 divided by the number of terms used which in this case was 10,000.