Why is the sum over all positive integers equal to -1/12? 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?
 A: Usually we do not assign a value to divergent series. The assertion that these kinds of series are divergent means that these kinds of series do not have sums. Now if we want to assign a sum (in some sense) to divergent series, what these sums "should" be and how they should be calculated?
Obviously we are going out of customary mathematics. We are extending it to "illegal" cases, and it is not a surprise that we get odd results.
G.H. Hardy's "Divergent Series" is a good reference. You can find more discussion on this subject there. 
I also found this video very helpful.

A: Basically, the video is very disingenuous as they never define what they mean by "=."
This series does not converge to -1/12, period. Now, the result does have meaning, but it is not literally that the sum of all naturals is -1/12. The methods they use to show the equality are invalid under the normal meanings of series convergence.
What makes me dislike this video is when the people explaining it essentially say it is wrong to say that this sum tends to infinity. This is not true. It tends to infinity under normal definitions. They are the ones using the new rules which they did not explain to the viewer.
This is how you get lots of shares and likes on YouTube.
