It's difficult to provide the proof here (my personal preference is the proof by integration) but the sum is 1/2. Not my favorite area of Mathematics, but that's one of the more interesting series.
As far as pi, I think it depends on how many digits you would need to see to consider it another pi. Since pi is irrational, it could never contain the fullness of itself.
I had a discussion in my history of math course recently where another student accused me of "faith-based mathematics" and said that accepting irrational numbers as being real is the worst thing we've done to modern mathematics because there's no evidence they actually exist. I told him good luck with whatever he's planning to do with half of modern mathematics in the trash bin.
:-D