*You can't do things infinite times* If we imagine the addition as a real-world activity that takes time to do -- a second to write the new sum on a piece of paper, a fraction of a microsecond for a processor to encode it in memory -- then the Sun will go out before you get 2
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Without writing out the proof, you see what kind of dirty trick I'm pulling here, right? For each positive fraction with an odd denominator, I'm "borrowing" an "extra" negative fraction with an even denominator from "the future"
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(Dug up a more in-depth discussion here http://larryriddle.agnesscott.org/series/rearrang.pdf …) If this were a finite set of numbers, then I would eventually run out of "future numbers" to borrow from, and this trick to make the sum constantly be lower wouldn't work, and the sums would come out the same
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But it's an *infinite* series, so I *never* run out And so the same set of numbers rearranged converges to 1/2 the original sum This is why doing things an infinite number of times is, again, a filthy lie
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(Riemann proved you can rearrange this series to converge on ANY sum, at all, or to not converge and instead spit out ∞ or -∞ Which is one of those things mathematicians do that really pisses people off)
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Anyway There are series that are absolutely convergent, and series that are conditionally convergent, and series that are divergent The series 1-1+1-1+1... doesn't converge, it doesn't "get closer" to anything the longer you do it, it just flips from 1 to 0
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So does 1-1+1-1+1... equal anything at all? According to the method people used when they started talking about this ("classical summation"), no, absolutely not There is no way to rearrange this series so it converges on anything, it's a divergent series
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A Norwegian guy named Abel (a deeply weird, extremely smart dude who invented all kinds of new math before he died of TB at the age of 26) was like "Sure you can, it's 1/2" You probably looked at that and said "Yeah it's 1/2", if you thought about it at all
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The thing is in math you are *completely allowed* to say "Okay the rules say you can't do it but it looks like it should be 1/2 so I'm gonna say it is" You just have to go on to explain to everybody what that *implies* if you decide to do it, which is the hard part
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I.e. by the delta-epsilon definition of a limit, which is "It has to get closer to the number every time you do the next thing", 1-1+1-1 by definition is not approaching any limit You do it once, it's 1, you do it again, it's 0, then 1, then 0
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The "distance" between each new sum and the answer 1/2 is exactly the same every time, you *never* get closer A so-called Abelian summation method involves using the idea of *averaging* instead of the idea of *limits*
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Does that make sense? Should that be allowed? The authorities haven't come to a moral conclusion on the matter, but hey it's fun and cool prizes come out when you do
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(Philosophically, taking an average of an infinite number of sums is worse than adding numbers together an infinite number of times, because it means you have to "divide by infinity", and that's a big can of worms Which Abel happily opened)
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It goes on from there Every new bleeding-edge summation method has more scary problems with it -- or, rather, it lacks certain properties that were assumed under classical summation Abelian summation does not "work" in many ways, but we accept that
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It "works" in other ways and that's okay Cesaro summation and Abelian summation are two methods that give you 1/2 for a "mildly divergent" series like Grandi's series (1-1+1-1)
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Abelian summation is slightly more "powerful" and can also handle weirder cases like 1-2+3-4+5, which according to him = 1/4
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If that bothers you -- "The number keeps *increasing* in absolute value, how can I go up to 2 then down to -2 then up to 3 then down to -3 and then at the very end of it get 1/4?!" -- well, you're in good company, Abel summation is said to have brought mathematicians to blows
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You see at every step we're leaving a little more of "common sense" math as a child would learn in grade school behind, we're redefining the meaning of the symbols "+", "..." and especially "=" a little more to say something else And that's fine
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And that's how we get to the real weirdness, where "1+2+3+4...=-1/12" This is a sum that does not "work" at all with Abel's summation method, it contains no "oscillation" (the switching from + to -) that was the secret sauce to make the 1-1+1 and the 1-2+3 stuff work
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*Any* answer that's actually a number is going to be wildly counterintuitive But asking "Okay, but what happens if you try it" is how math works The dude who tried it was Ramanujan, a real weirdo whose life story is tremendously inspiring, and who was kind of an awesome troll
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The method Ramanujan used to prove "1+2+3+4...=-1/12" is summarized in this famous YouTube people, which made a lot of people very angry and was widely considered a bad movehttps://www.youtube.com/watch?v=w-I6XTVZXww&t=303s …
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It's important to note that this method, stated in the form the guy gives in this video, is wrong He's doing multiple things you're just not allowed to do Ramanujan liked that kind of shit
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I.e. if you naively just assume everything he's doing -- adding infinite series to each other, multiplying a finite term across an infinite series, etc. -- is allowed in every circumstance, you get contradictions
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Trying to build a summation method this way for a diverging series 1+2+3+4... does not work, the way 1+2+3+4... diverges makes it neither "linear" nor "stable" I.e. if I stick an extra zero in there ("0+1+2+3+4..."), I can use his method to prove 1=0
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And 1=0 is generally frowned upon, people don't like it, it's dogs and cats living together As a great philosopher once said "If SOMETHING is the SAME THING as NOTHING then YOU COULD HAVE ANYTHING You can't just have anything, you've got to keep the riffraff out"
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And yet it's not *arbitrary* Ramanujan's "dirty trick" isn't just bullshit, I can't actually use it to prove anything I want If you do the specific thing he's doing the way he's doing it, consistently, every time, you get consistent answers
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Whatever the hell he's doing, even if it isn't "adding up numbers" in any recognizable way as we normally do it, it is a *real thing* and it always gives you the answer "-1/12"
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Ramanujan famously told Hardy in his letter about this "Please read to the end of the page before sending me to the insane asylum", and Hardy let himself get infected with the brainworms before calling the authorities in time to stop it, and now here we are today
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Formalizing the method Ramanujan used here -- explicitly stating what rules actually define the things Ramanujan was doing so you can't just do anything and get 1=0 -- created the method known as "Ramanujan summation"
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Which ended up being a special case of the regularization of the Riemann zeta function, a form of analytic continuation (which is a phrase for "doing things you're not allowed to do") Which the guys who made the above video go into detail about herehttps://www.youtube.com/watch?v=E-d9mgo8FGk …
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I fully admit I do not really understand what they are talking about, and that my brain plasticity at my age is already probably too low to ever actually feel like I have the energy to really get into it But it's a real thing, and it works
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