Why is the sum over all positive integers equal to -1/12? [duplicate]

Recently, sources for mathematical infotainment, for example numberphile, have given some information on the interpretation of divergent series as real numbers, for example

$\sum_{i=0}^\infty i = -{1 \over 12}$

This equation in particular is said to have some importance in modern physics, but, being infotainment, there is not much detail beyond that.

As an IT Major, I am intrigued by the implications of this, and also the mathematical backgrounds, especially since this equality is also used in the expansion of the domain of the Riemann-Zeta function.

But how does this work? Where does this equation come from, and how can we think of it in more intuitive terms?

marked as duplicate by Matthew Conroy, Emily, Jean-Sébastien, mrf, Ian MateusApr 11 '14 at 18:58

• This has been asked at least twenty times before. Try looking for those questions. For what it's worth, the equality is false. – Git Gud Apr 11 '14 at 17:32
• @MatthewConroy wow, I wonder why this didn't come up in my search... Thanks for the link. – Andreas Grapentin Apr 11 '14 at 17:33
• look also this question: math.stackexchange.com/questions/735190/sum-of-divergent-series – Alessandro Codenotti Apr 11 '14 at 17:34
• @AndreasGrapentin It's a little hard to search for, I think. For instance, searching for "-1/12" yields lot of stuff that doesn't have -1/12 in it. – Matthew Conroy Apr 11 '14 at 17:36
• Yes, this question has already been asked many times here, an unfortunately some people are pretty hostile to it now, but no worries. Definitely check out their answers! – abnry Apr 11 '14 at 18:04