This is called "absolute convergence" If the numbers change sign, you have "conditional convergence" Paradoxically, for people used to doing things a finite number of times, it suddenly matters how the numbers are arranged
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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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It doesn't map onto the real world in a familiar way very well, at all But it wasn't meant to Counting things up infinitely isn't something you can do in the real world AT ALL either, remember?
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And yet, bizarrely, this result is meaningful in an applied context It comes up in quantum physics, where the real physical world gets extremely weird and we need to use weird math to talk about it
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It's part of the mathematical description of the Casimir effect, which has been experimentally verifiedhttps://en.wikipedia.org/wiki/Casimir_effect …
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The extremely broad metaphor the guy gives us for trying to imagine this is -- the actual positive numbers that we add up in 1+2+3+4... can't give us a real sum The number keeps getting bigger, it leads us only toward ∞, and ∞ isn't a number, so the sum just doesn't exist
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Analytic continuation is about saying "Okay, so if you take out everything that you're not allowed to do that absolutely by the rules cannot give you an answer, can you find something about it that *does* give you an answer"
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Paradoxical Zen koan shit Like Death in Discworld saying that the universe by definition is everything that exists and everywhere is inside it so you can't be outside it and standing outside it is a contradiction in terms But if you could, then from the outside it'd be blue
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-1/12 is a "residue", it's a "leftover piece" of 1+2+3+4..., the one bit that can be made to act like a number
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And that, in its way, is connected to the paradoxical thing the Casimir effect is, this thing where in a total vacuum where nothing should be exerting any force on anything, two metal plates very close together will attract each other
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