A strange derivation:
This sum 1+2+3+... = -1/12 is used in physics a lot believe it or not.
This can be rationalized sort of:
The Riemann zeta function is zeta(s) = 1/1^s + 1/2^s + 1/3^s + 1/4^s + ...
where x^s means "x to the s'th power", and s is a complex number.
This series converges when the real part of s is >1. This function can be analytically continued over the whole complex plane, except for s=1, and this analytic continuation is called the zeta function.
Define a function yeta in TWO complex variables by
yeta(s,z) = 1/1^s + z/2^s + z^2/3^s + z^3/4^s + ...
The yeta function converges for |z|<1, for all s.
Then we have the functional equation
yeta(s,z)- 2z/2^s yeta (s,z^2) = yeta (s, -z), for |z| < 1.
When Re(s) > 1, we can take the limit as z -> 1, to get the equation
zeta(s)(1 - 2/2^s) = lim (z -> 1) yeta (s, -z)
Since the zeta function can be analytically continued to the complex plane except for a pole at s=1, the RH side can be analytically continued as well.
We have yeta (-1, -z) = 1 - 2z + 3z^2 - 4z^3 + 5z^4 - ... = 1/(1+z)^2
Taking the limit as z -> 1, we get
lim (z -> 1) yeta(-1,-z)=1/4.
So if the analytic continuation of lim (z -> 1) yeta (s, -z) = lim (z -> 1) yeta (-1, -z) for s=1, we have
(-3)zeta(-1) = 1/4, or zeta(-1) = -1/12.
infinitely many 1's cannot possibly sum to something less than 1 (let alone something negative).
You can't sum infinitely many 1's in the usual sense of addition. You can add finitely many numbers together, but to define the sum of infinitely many numbers involves some definition of a limit. Without that definition of a limit, there is nothing to say what the sum "should" be.
Someone I know sent a mathematical musing to me. I told him if you have an infinite sum that is conditionally convergent but not absolutely convergent, you can rearrange the terms to come up with any limit you like.
He objected "But addition is commutative!" So he figured infinite sums should be rearrangeable without changing the value. Not true.
Another example: a finite sum of rational numbers is rational. An infinite sum of rational numbers can be irrational!
So why should an infinite sum of positive numbers be positive?
I hope physicists are indeed using a harmless shorthand.
It seems to be an intuitive shorthand such that they're using the zeta function without making it explicit. Complex numbers are very important in physics, Roger Penrose has wondered a lot about that.
ps The zeta function does have a pole at s=1, which is a bit ironic since 1 + 1/2 + 1/3 + 1/4 + ... looks "less divergent" than 1+2+3+...
But the sum zeta(s) = 1/1^s + 1/2^s + ... converges for Re(s)>1, and tends to infinity as you approach s=1. So the zeta function has to have a singularity at s=1.
I'm sorry to say that I never learned enough about analytic continuation or about the zeta function to say anything intelligent about either one. And my complex analysis books are in storage. Anyway, I suspect there would be points at which I would need someone to help me out, were I to dig out my books or to look the subjects up online.
Fancy footwork can be impressive--but not when it gives an obviously false result. When such a thing happens, it's an indicator that something has gone wrong in the derivation.
What I wrote following the video is an attempt at rationalizing their derivation.