What if you sat down and added up:
1+2+3+4+5…
all the way to infinity. What would you get? Most people would say – well – infinity.
What if I told you it wasn’t infinity – but -1/12. Hogwash you’d say!
But here he is proving it, and he claims the result is used all over in string theory and astronomical physics.

How? Well, it has to do with how you think about and sum infinite series.

Spoilers:
I remember a prof telling me that depending on how you interpret infinity – you can get systems in which parallel lines converge or diverge or stay parallel at infinity.  It didn’t make a lot of sense at the time, but now you see a real case.

You start to enter the rabbit hole when he proves the Grandi’s Series = 1/2:
1-1+1-1+1…=1/2.

Here’s a link to all those explanations
http://periodicvideos.blogspot.co.uk/2014/01/thanks.html
But how does it really work?  TerryTao goes into tremendous detail of how the summation works and why the Grandi series can sum 1-1+1-1+1… even though it does NOT converge in the classic sense. But that’s the difference between applied mathematics/physics and theoretical.  In applied fields – you really DON’T have infinite things summing – you have real things – and it makes modeled results…unexpected.  Amazingly enough – we had models for these systesms before we even knew of real things that displayed the properties.