Is this true? I watched the Numberphile video but it still baffles

First, this is literally the most obnoxious maths mene and notation abuse ever devised by humanity.
Second, Numberphile's "proof" mistreats divergent series and basically isn't a proof at all.
You should check out Riemann's functional equation and how it's used to construct Riemann's zeta analytical continuation.

>>8585770
https://www.youtube.com/watch?v=sD0NjbwqlYw

The video is long, but very well made and explain all.
To sum up :
In the domain of real, 1 + 2 + 3 + ... does not equal -1/12. The main argument used by numberphile (that 1 - 1 + 1 - ... = 1/2) is false, because you can just arbitrarily change the order of the terms of a sum that does not converge.

However, the function defined by [math]f(s) = 1 + 1/(2^s) + 1/(3^s) + ... [/math], more known as the Zeta function, can be artificially prolonged outside of where it exists.

When you uses this prolongation at the point (-1), then you have [math]f(-1) = -1/12[/math]

That does not mean : [math] 1 + 2 + ... = -1/12[/math] ; the sum [math] 1 + 2 + ... [/math] does not exist, mathematically speaking. It is infinite. However, we can artificially extends f in a way that f(-1) = -1/12.

>>8585801
>notation abuse
Classical analysis shouldn't have the monopoly on those symbols

It's notation out of the context of more relevant math, but there's no abuse. If I use [math] \omega [/math] to denote both a rotation vector and it's magnitude, then that's abuse of notion. An equation from some context that also has an interpretation in a more common context isn't abuse..

>>8585837
I'd argue the + sign should be used with decimals in traditional sense only, but k lad

File: string.theory2.jpg (50KB, 400x400px) Image search: [Google]

50KB, 400x400px

Does anyone know when or why it is physically acceptable to use an analytic continuation. To the naive it might appear like a logical fallacy that an extension of a domain would have a physical meaning.

>>8585909
Because it's unique.

>>8585821
Transitivity may not hold, but associativity

>>8585770
The number is so big that it rolls over past infinity and down through the negatives.

>>8585909
b/c it aligns with experiment

Here's your explanation, OP.
Let [math]N \,\in\, \mathbf N[/math].
[eqn]\sum_{n \,=\, 0}^N n \,=\, \frac{N \,\left(N \,+\, 1\right)}{2} \,\xrightarrow[N \,\rightarrow\, \infty]{}\, \infty[/eqn]
Therefore [math]\sum_{n \,=\, 0}^\infty n \,=\, \infty[/math].

It's easy to confuse yourself with this shit but it's quite simple.

All that the ramanujan summation stuff, cutoff and zeta regularization does, is look at the smoothed curve at x = 0.
What sums usually do is look at the value as x->inf.

It's just a unique value you can assign to a sum, really they have many such values.

https://en.wikipedia.org/wiki/File:Sum1234Summary.svg