The circumference of a circle has a simple formula $ c = 2 r \pi $, but there is no simple formula for circumference of an ellipse with major and minor axes of a and b. There are approximations, including the not very close $ c=(a+b)\pi$, but this infinite series $$ h = \frac{(a^2)-(b^2)}{(a^2)+(b^2)}$$
$$p=\pi (a+b)\sum_{i=0}^\infty \left(\frac{(2n-3)!!}{2^n n!}\right)^2 h^n$$ gave me an idea.
In some ways you can think of the summation part as a correction factor to $\pi$. Let's say that we pick a particular ellipse with a=5 and b=1. Then $\pi$ is a few percent too low to make $ c=(a+b)\pi$ work, so we need a different constant to multiply times (a+b).
Can we make a new infinite series not containing the letter $\pi$ for this very specific ellipse? I know this is kind of a silly task since every specific shaped ellipse would need its own infinite series, but it would be useful for a talk I'm to give.
My thought was to Cauchy product rule to multiply the summation part of the formula above with one of the infinite series for $\pi$, but it turns out that is beyond my skills. Can someone help?
Thank you in advance.