What is the limit of $\frac{\prod\mathrm{Odd}}{\prod\mathrm{Even}}?$ Does $\pi$ show up here? 
What is this limit
  $$
  \frac{1\times3\times5\times\cdots}{2\times4\times6\times8\times\cdots} = \lim_{n \rightarrow \infty}\prod_{i=1}^{n}\frac{(2i-1)}{2i}
$$

I remember that it was something involving $\pi$.
How can I compute it?

In addition; how can I compute it's sum series limit?

Here's a formula which I found embedded in an old C program. I don't know where this comes from, but it converges to Pi very quickly, about 16 correct digits in just 22 iterations:
$\pi = \sum_{i=0}^{\infty}{
\frac{6(\prod{2j-1})}
{(\prod{2j})(2i+1)(2^{2i+1})}}$
(Each product is for j going from 1 to i. When i is 0, the products are empty, equivalent to 1/1. When i is 1, the products are 1/2. When i is 2, the products are 3/4. Etc.)
The limit of the quotient of the products is 0, but the limit of the sum as I give it above is exactly equal to $\pi$.
As for the limit of the sum of the quotients of the products, it diverges to +$\infty$. 
