Apologies if I'm in the wrong forum.
The ratio of two successive terms in the Fibonacci series converges on the golden ratio. Are there equivalent number series that evaluate pi or e?
EDIT: way too slow!
I'm not sure if it's what you want, but the sum of alternating series of reciprocals of odd numbers is $\displaystyle \frac{\pi}{4}$.
Also, one of the definition of $\displaystyle e$ is the sum of the series of reciprocal factorial numbers. See here, btw.