ASTOUNDING: 1 + 2 + 3 + 4 + 5 + ... = -1/12

Watch this entertaining Numberphile video about this topic: [VIDEO] Apparently it's used in many areas in Physics.

this infinite sum was featured in a numberphile video a few days ago with a entraining and simple "proof": [VIDEO] and a more advanced explanation & proof in the bonus-video: [LINK] you should definitely subscribe to numberphile ( [LINK] ) and their sister channel computerphile ( [LINK] ) on youtube :)

I think the quality of numberphile videos vary quite considerably. The one you linked is a rather honest explanation of the idea of comparing the size of sets through bijections and a good explanation of Cantors diagonal argument. As a mathematician I'm quite fine with it. The one linked in the submission is more of an obvious statement hidden by obscure use of language. Slightly silly, but not very bad. The worst offender with "magic" is the one with the Riemann zeta function [1], which went viral a while ago. The problem here is that they get people started off on the wrong foot, confusing them with wrong arguments and hidden definitions. Now, if people really want to understand why this can be made meaningful, you first have to explain that the better portion of the video is absolutely wrong, and only then can you explain what is actually going on. [1] [VIDEO]

I love the idea, but I am concerned that if run by and for physicists, it will perpetuate the lack of rigor and "shorthands" physicists use to abuse mathematics [1]. I would propose an additional feature: that entries be highlighted in an additional way if they are mathematically false as written (or misleadingly overloaded, or just a huge abuse). [1] e.g. the sum of natural numbers being -1/12, perpetuated with reckless abandon in this video [VIDEO] to the dismay of mathematicians everywhere. I also have personal gripes with the way physicists use delta functions, and hear many physicists say (about some topic in mathematics), "I don't want to actually do the mathematics here, I just want to get intuition about it."

There are a lot. I thought this one was really funny [0]. This one I found very interesting [1]. Somewhat different, is the Hello Internet [2] podcast, which is by Grey and Brady Haran who you might know from the Numberphile youtube channel [3]. It's basically the 2 of them chatting about random stuff, but I find it very entertaining. [0] [LINK] [1] [LINK] [2] [LINK] [3] [VIDEO]

I learned about it through this Numberphile video: [VIDEO] There's also a nice write-up from Terry Tao: [LINK]

Few days back I stumbled upon a video about Ramanujan [0] and the: 1 + 2 + 3 + 4 ..(to infinity) = -1/12.(Turns out the proof in the video is incorrect or at least not concise enough, because you can't do that with divergent series). So that got me curious and I wanted to understand what's going on. How can 1 + 2 .. etc equal to -1/12 and not infinity. So I went on small binge watching videos to get an intuition and understanding about this. This led me to Riemann Hypothesis and Euler's identities and complex numbers etc. Now I have a list of books that I want to skim/read to get a historical context around these problems. I don't take any notes just yet, because I don't know what is important and what is not, I just jump into the stuff and hope that when I come out of the other end I will at least have a understanding why it equals -1/12. It's like panning out gold, there might be some gold nuggets. And maybe, just by pure chance, I come up with some ridiculous ideas how to disprove Riemann Hypothesis ;P (I think I have to revisit Gödel, Escher, Bach and Gödel's Incompletness Theorems to prove that it's not provable). - [0] [VIDEO] EDIT: Sorry to all facepalming mathematicians who happen to read this comment.

It's referencing a theorem from string theory, as discussed in this Numberphile video: [VIDEO] The catch is that this isn't valid in what we consider standard mathematics, and you can find many discussions of this online, but this one is fairly short and straightforward: [LINK]

This is a good video explaining it: [VIDEO] Although it relies on a particular interpretation of the left side of the equation, it's used in some areas of physics.

You are right and I am glad I didn't write 'because Maths'. The sum you're quoting from is monotonically growing but at an ever shrinking clip. However, the series does not describe the kind of growth that our economy seemingly needs and our politicians like to promise, nor does it AFAICS describe the OP's idea of continued population growth given his recommendation to shun shrinking populations with fertility below replacement rate; theirs is a constant or in the midterm increasing growth and notwithstanding [minority opinions]( [VIDEO] ) claiming that 1 + 2 + 3 + 4 + 5 + ... equals -1/12 such a series cannot have a finite sum, hence is physically unfeasible to represent.

I look forward to every Numberphile (and Sixty Symbols and Deep Sky and ...) video. I am pleased to hear that Brady's questions are off-the-cuff. They're so on-point that I thought they may have been discussed with the guest beforehand. The 1+2+3+4+... = -1/12 video still seems a bit of a cheat, though! [VIDEO]