Recently there's been a lot of buzz created by this video http://www.youtube.com/watch?v=w-I6XTVZXww which states and goes on to prove
$$1 + 2 + 3 + ... = -\frac{1}{12} $$
I know that the above series is actually $\zeta(-1)$ where $\zeta$ is the Riemann zeta function. For someone who is not much into higher and advanced mathematics, what meaning does this equation hold? The video explain that this is a very useful result used in string theory in Physics and the basic premise which is used to prove it is
$$1-1+1-1+... = \frac{1}{2} $$ which doesn't seem quite right and which, in fact, leads to contradictions as explained here on Quora http://www.quora.com/Mathematics/Theoretically-speaking-how-can-the-sum-of-all-positive-integers-be-1-12/answer/David-Joyce-11. So, my question is is the proof shown in the video meaningless and only serve the purpose of sensationalizing by wrongly interpreting and misrepresenting mathematical results?