For a visual understanding, this video by math animator 3Blue1Brown is superb. (Its where I got the picture from). And if you are brave then learn from the greatest mathematician alive about the meaning of these strange sums in this article
The incorrect proof
The mistake comes from assuming convergence on a sum, and then applying rules which are only justified if a sum does converge. The mistake in the proof given, is when it writes:
1 + 2 + 3 + …. = C
The placement of that C on the end is an assumption. It assumes that this sum has a well-defined value, on which standard operations (addition, subtraction, mulitplication, division) are then defined. But this is obviously untrue.
While popular articles such as this one acknowledge the lack of convergence, as soon as you acknowledge this, the rest of the proof loses all its meaning. It is as profound as saying 1 + infinity = infinity = infinity + infinity, so 1 = infinity. (i.e. a load of garbage).
Or, as Terence Tao (my hero!) puts it:
Clearly, these formulae do not make sense if one stays within the traditional way to evaluate infinite series, and so it seems that one is forced to use the somewhat unintuitive analytic continuation interpretation of such sums to make these formulae rigorous.
P.S. this is not to single out the author of this article, as the misconception is now so widespread that unless you have an eye for pure mathematics this will slip past you. When I first saw the sum and proof, I was star-struck too!
What It Actually Means
There is a function called the Riemann-Zeta function.
Zeta(s) = 1^(s) + 2^(s) + ….
For instance, Zeta(-2) = 1 + 1/4 + 1/9 + 1/16 +….
And Zeta(-3) = 1 + 1/8 + 1/27 + ….
This can be extended to non integers, for instance, zeta(pi) converges, and can also be extended to complex numbers — numbers of the form x+ iy.
If you put in Zeta(1), the function isn’t defined. This is because 1 + 2 + 3 + 4 + … does not converge.
However, the function, when plotted in the complex plane, is simply so aesthetic when extended to include these numbers. It is literally begging for this to be done, I promise. Just look at the picture at the beginning, before and after analytic continuation.
The equality comes from assigning a value to the Riemann-Zeta function, but one which is more derived from our aesthetic expectations of what the function should look like. and it turns out for complex functions there’s only one ‘nice’ way to define the analytic continuation. And so it happens that the value assigned at s = 1 is… (drumroll) -1/12!
But this value has been assigned as the meaning: at s = 1, the Riemann-Zeta function is no longer 1^(s) + 2^(s) + 3^(s) + …., but it is whatever we chose it to be.
(The technical name for this is analytical continuation, where a function is extended beyond where its meaning is strictly defined in order to create some nice properties. It is worth noting that analytic continuation has real uses in solving problems)
In fact, in Terence Tao’s article, he explains a way of creating a meaning for the sum using only real variable calculus. But that’s less standard.
Beauty, not truth!
The killer combination is that the sum is so striking, and the supposed proof so elegant, that this statement soon lost sight of its origin.
But now you know. And if you want to understand it all better, see this video by 3Blue1Brown
The author is a mathematics enthusiast, with a special love of real analysis.