Start with this infinite sum: 1 - 1 + 1 - 1 + 1 …. Depending how many summands we consider, the sum is either 1 or 0. So let’s take the average of both values.

This seems to be a not unreasonable choice.

Now look at the sum $S_2 = 1 - 2 + 3 - 4 + 5 \dots$ what’s the value of it?

$2S_2 = S_1$ gives $S_2 = \frac{S_1}{2}$ and therefore we get this less intuitive result:

The next sum is $S_3 = 1 + 2 + 3 + 4 + 5 \dots$. We find that

$S_3-S_2 = 4S_3$ gives $S_3=-\frac{S_2}{3}$ and we get this totally crazy result:

The sum of all positive integers is equal to minus one twelfth!? This is clearly utter nonsense. It turns out that it’s not. It’s even used in physics, e.g. in string theory.

The problem is caused by the first step, by assigning $\frac{1}{2}$ to the first sum. In one part of mathematics, this is not correct as the sum does not converge. But with analytic continuation, it’s valid to assign a value to an expression that otherwise has none. The results that follow from this assignation are mind-blowing but they apparently make sense. Crazy stuff!

Thanks to Numberphile for producing such cool videos.