natural numbers - 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?

Answer

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.