# 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
• @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