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u/LuxDeorum Nov 12 '20
If you do a silly thing like say that the sequence 1,0,1,0,1,0... converges to .5 then you can show that the sum of pos integers converges to a negative number
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Nov 13 '20
Silly is a strong word considering there are algebraic and analytical proofs utilizing taylor series expansions that show that result. Additionally, it can be argued that the analytic continuation of the zeta function implies this value as well.
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u/Dark_Ruler Nov 13 '20
A convergent series converges to one value. Divergent series does not converge to one value. So the summation of natural numbers does not converge to single value. So silly is correct word.
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u/Traveleravi Nov 13 '20
Things are true or not based on the assumption they are based on. Calling things silly is rather silly. But making assumptions and definitions and then proving something based on those assumptions and definitions is math, regardless of how unusual it seems
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Nov 13 '20 edited Mar 19 '21
[deleted]
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u/cavalryyy Nov 13 '20
Not to be crass, but this is completely wrong. Starting from assumptions and new definitions and seeing the implications of those definitions is the essence of math. For example, we can perfectly well say “define 1/0=0”. This is “fine” in the sense that anyone can make this definition. It doesn’t “illegitimize” proofs you do using that definition, those proofs just don’t say anything about arithmetic using the standard definition of division.
The same thing goes for defining infinite sums. You don’t have to define the sum “1+2+3+... = infinity”. Perhaps that’s the most natural definition, but I can just as easily say “define the sum 1+2+3+... = 5” and see what happens. Of course, it is exceedingly unlikely that definition will lead anywhere interesting, but there’s nothing stopping me from making that definition, and proofs I do using it are still valid math, they just don’t make any claims about sums using your definition. It just so happens that someone defined something (the analytic continuation of the zeta function) in such a way that we can, if we want, consider things like the sum of all natural numbers to equal -1/12. That fact doesn’t say anything about infinite sums using the standard definition, but some people care about the analytic continuation of the zeta function, and therefore by extension they’re interested in its applications. Does that make the sentence “the sum of all natural numbers is -1/12” a silly one? Maybe to you, but not to them. And regardless of which side of the aisle you’re on, as long as you’re making valid deductions from your assumptions, you’re both doing math.
Unrelated, but someone has horribly misled you about what induction is.
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u/LuxDeorum Nov 13 '20
Right but if theres a reason we add the word "formal" when talking about formal power series. Saying the sequence converges to .5 is silly because the sequence doesnt converge.
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Nov 13 '20
Ok ok it doesnt converge, but like, if it DID...
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u/LuxDeorum Nov 13 '20
Lol yeah that's why it's silly my dude.
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Nov 13 '20
cries in -1/12
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Nov 13 '20
I am with you and ramanujan
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Nov 13 '20
Thanks friend, idk why tf im getting downvoted so hard for stating mathematical facts
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u/Shabam999 Nov 13 '20
Yeah no you're definitely right. These sums make as much sense as the square root of -1 or having different sizes of infinities. There's multiple rigorous ways to define that sum and they all give the same answer. To be completely fair though, the way it is introduced to most people (i.e. using the sum tricks in the numberphile video) isn't rigorous but it does get to the right answer. I was planning on commenting earlier but there's honestly no point. I think one of the biggest difference between math people and non math people is that if their intuition disagrees with the math, they think the math is wrong and not their intuition and the opposite holds true for math people.
And it's not like the numberphile people don't know the proper way to do it. I found this video on an alternate channel but the problem is that this stuff is quite complicated and not something you can learn from a 20min youtube video and most people don't really want to learn, they just want to feel smart/like they learned something.
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u/LuxDeorum Nov 16 '20
You're promoting a pretty toxic perspective that doesn't belong in the "math people" community, and to make it worse is that you're just wrong.
Abel and Cesaro Summation, the first methods to supply the 1/2 value for the grandi series are both members of a general family of summation methods which can be described as follows. Each method is given by an array A(m,n), and evaluation works by taking the sequence you want to evaluate, S_n, forming a new sequence T_m= \Sum_n A(m,n)*S_n and then taking the limit of T_m as the value of your limit. Of course for this to be meaningful, we need that if S_n is already convergent, the the limit of T_m will converge to the same value. This is guaranteed by these conditions on the array A(m,n):
1: For every m, \Sum_n |A(m,n)| < inf 2: Lim_m->inf \Sum_n A(m,n) = 1 3: Lim_m->inf A(m,n) =0
A good exercise would be proving this.
Cesaro summation corresponds to A(m,n) = 1/m if n=<m else 0
Another good exercise is seeing how Abel summation fits into this family (hint, let m be a real number in (0,1), and take it's limits to 1 instead of infinity).
Anyway, what we have are a family of arrays A(m,n) which correspond to rigorous defined methods for assigning values to series which are not cauchy convergent. Plenty of these do not deliver the estimate that you claim is universal. For example, let A(m,n) = 1/mn. You may check that this array satisfies the conditions I gave above, and you may also check that it calculates for the series (1-1+1-1...) a value of 1.
These neat mathematics aside, the point is made moot because these methods of summation were developed prior to the publish of and consensus on Cauchy's definitions on limits and convergence, which categorically rejects the notion that these sums have well defined values -- Within the context in which Cauchy's definitions apply --. This context includes standard analysis and absolutely includes the normative understanding of what it means to add numbers. Every single undergraduate analysis professor in the country would accept "The limit of a strictly positive sequence is nonnegative" as disproof the Ramanujan sum. As an analogy I would borrow your mention of the number i. It is certainly the case that there is no square root of -1, when those numbers are understood in the context of quantity or as scaling operators. A line cannot be drawn which is i long, and there is no scaling operator which inverts the direction of a vector when applied twice. Of course, you would realize two quarter rotations would do and you're well on your way to complex analysis. Multiplication of real and imaginary numbers are conceptually distinct, and it would be goshdarn silly, to tell a person who has no idea what imaginary numbers actually describe that there is a scaling factor that flips a vector backward when applied exactly twice and anyone who doesn't think so is intellectually lazy.
I dug out old notes and worked out the counterexample for you so that when you see this you'll A: see that you were wrong about something, and maybe be more open to considering how your arrogant attitude is hurting our community and B: get to read some neat math that maybe you haven't seen before, because that's what the math community really is about.
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Nov 13 '20
Let them downvote It is due to lack of their mathematical knowledge Had it been r/math this would not have been the case
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u/5059 Nov 13 '20
It’s delicious but it takes forever to prepare and you have to use some unconventional ingredients
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u/_saiya_ Nov 13 '20
I read a proof where this was equal to -1/ 8 and it was a meme really with thanos saying impossible but here's the proof.
S = 1 + (2 + 3 + 4) + (5 + 6 + 7) + (8 + 9 + 10) ....
S = 1 + 9 + 18 + 27 + ...
S = 1 + 9 ( 1 + 2 + 3 + ... )
S = 1 + 9S
S = -1/8
And thats pretty neat proof imo. The -1/12 is also along similar lines where they proove sum of a bunch of series prior to these and using that they derive this. Also in math we do use lim x--->∞ sin x as 0 based on similar approach although idk the idea of infinity seems crazy especially in 2d plane as in in complex nos and vectors because there are so many infinities. It's a bit crazy 🤪
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u/aortm Nov 13 '20
Okay so.. Every possible summation of that series has either been infinite, or -1/12. Seemed like -1/12 was a good answer, other than that non-useful infinity.
How did -1/8 pop out?
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u/_saiya_ Nov 13 '20 edited Nov 13 '20
You have the proof above about how it -1/8 pop out.
The sum tends to infinity if the series is finite somehow, like it has a definite end somewhere so if we add upto some number the sum will be infinity.
But if the series truly continues to infinity then it uses the property that any number of addition or subtraction of numbers wouldn't really change the units in series and hence this arrives. For eg.if you have 1 to 100 numbers. So if you add 2 neighbouring nos to form new series it'll have 50 numbers. But such things wouldn't be true for infinities because ∞/2 is ∞. Or any algebraic operation really wouldn't change the count.
Like I'll give another example. Let's say there are n numbers from 0 to 1. Let the number itself be x.
Corresponding to every x in 0 to 1 there is a number x +1 in 1 to 2. * So now there are n numbers from 0 to 1 and n numbers from 1 to 2. *
Now for every x number between 0 to 1 there is a number 1/x between 1 to ∞. * So there are n numbers between 1 to ∞ now. *
If there are n numbers from 1 to 2 and there are n numbers from 1 to ∞ this implies there are no numbers from 2 to ∞.
So the only solution to this conjuncture is if n is infinite. So there are infinite numbers between any 2 numbers. It's difficult to comprehend in some ways the concept of infinities but seems we've already accepted without much thought in numbers ;-)
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u/aortm Nov 13 '20
No you don't get me. All of the various ways to get any finite value from the infinite sum has been -1/12. I thought it was a given that this sum is somehow associated with -1/12 already.
now you've shown me some derivation/proof its -1/8. How do mathematicians handle this?
-1/12 is clearly not equal to -1/8
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u/_saiya_ Nov 13 '20
I really don't know how they handle any of this. I'm not even studying maths formally. I'm a civil engineering major. I like to read math and so ik this. But haven't really asked some mathematicians about this. But clearly math sort of breaks down if we tend to infinity because additions multiplication and other operations etc don't work. We somehow arrive at a number but I'm not even sure if I can accept that. A negative fraction :-)
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u/Rufus_Reddit Nov 13 '20
... How do mathematicians handle this?
One easy way is to just say that the talk about -1/12 and the talk about -1/8 is all bunk in the first place. Then there's no problem at all.
The Mathologer videos on this topic are pretty good. https://www.youtube.com/watch?v=YuIIjLr6vUA
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u/LuxDeorum Nov 16 '20
The -1/12 sum is saying something different than you think, using the same symbols. Mathematics is language that describes certain ideas, and sometimes a sentence is patently untrue taken a certain way, and profound taken another way. The proof saiya gave is false because the kind of manipulation he did is only allowed when the sum is "absolutely convergent". The -1/12 proof is not a proof that the sum converges to -1/12, but something else entirely.
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u/hardstuck_silver1 Nov 13 '20
The joke? The sum of all positive numbers = -1/12
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u/sci-goo Nov 13 '20
That's not a joke, but the "sum" talks here is also no longer the serial sum.
It's an extension to the serial sum, and it is proven to be useful (in physics). Like the imaginary number i, if you managed to accept it, you can also manage to accept 1+2+3+... = -1/12 (R). The (R) denotes a Ramanujan sum.
Note each time math extends its realm and method, it must be concise within the old system and is useful. Mathematicians don't care about inventions of no use.
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u/LuxDeorum Nov 16 '20
"extension" is a bit of a misnomer here. When someone writes the equation 1+2+3+.... = -1/12 they are not really talking about sums in the sense that we understand them at all. I mean it's totally impossible for a sum of positive numbers to be negative when sum is meant in the usual way and the numbers are the numbers we usually mean. What's really happening here is that this equation is expressing something distinct from our normal interpretation of sums using the same symbols (for good reason). For example the number i is only dubious when you imagine real numbers to exist purely as quantities, and of course you could never have i number of objects or draw a line that is i long. Rather, it happens that very naturally numbers can be thought of as representing geometric objects or operations, and in this context is very natural to include i. In fact imagine real numbers as just the lengths of segments rooted at 0 on a number line (negatives going one way and positives the other, and multiplying by a number just meant scaling the segment to the appropriate size and direction. In this case multiplying by -1 would just mean flipping the direction of the segment without affecting its length. So if I were to ask is there a geometric operation on the segment which when applied twice does the same thing that multiplying by -1 does, the answer of clearly yes, a 1/4 rotation! The point when talking about i, you have extended your number system's capacity to describe a certain kind of thing, but not another. A cheeky image: your company is losing 1m$ a year per factory they have running, and your boss asks can we solve this problem by building more factories? you say "of course, if we scale up total production by a factor of i for the next two years, our situation will be completely turned around!"
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u/sci-goo Nov 16 '20 edited Nov 16 '20
Ramanujan sum is a special case that works on divergent series. In the same theory, another special case would just work on normal convergent series. Yes, the theory is an extension to ordinary serial sum. It is not an extension of how to calculate the sum, but the concept of sum itself.
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u/LuxDeorum Nov 16 '20
My point is that just as in extending the real numbers by the complex number we also lose the uniform interpretation of multiplication as scaling, so too in applying various schema to assign values to divergent series we lose something. I understand that what you mean we are extending is the concept of sum, I just mean to underline here that in doing so the 'sum' we are now using is substantially conceptually distinct from the sum of standard analysis.
If you're familiar with wheel algebras then you could ask yourself do you think a wheel algebra is just an extension of our traditional algebra to define division by zero?
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u/sci-goo Nov 16 '20
Unfortunately I'm not familiar with wheel theory, but I think our opinions are similar. I consider an "extension" would be any theory extended to new objects/areas that:
- The new theory is concise in the old axiomatic system (e.g. new axioms can be introduced but cannot contradict old ones).
- The new theory restricted to original objects replicates the unextended theory.
For example,
- Addition in R is an extension of addition in Q.
- General relativity is not an extension of Newtonian gravity as they are fundamentally different.
I don't think extending multiplication to complex numbers "loses" the interpretation of scaling... It simply gains other meanings when neither of the operands is real. When extending a method, the interpretation also extends. Specifically, the multiplication on C restricted to R has still the normal "scaling" interpretation. In fact, if you define "scaling" as "grabbing the multiplication neutral element and scale it to the multiplying value", the "scaling" interpretation retains.
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u/CROW_98 Nov 13 '20
This is the best explanation I have ever found on this topic
Really Interesting to see how it is related to Complex Analysis.
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u/pikachu2979 Nov 12 '20
Numberphile's video
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u/Chand_laBing Nov 13 '20
It's an entertaining video, but such a totally imprecise and misleading way of explaining it that it honestly becomes unhelpful.
Summability methods for divergent series are such dramatic generalizations of normal additions that they are essentially redefinitions entirely.
So, I wouldn't recommend at all for beginners to watch the Numberphile video; watch the Mathologer video instead or read the Brilliant article.
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u/Imugake Nov 13 '20
There are multiple ways in which the value for the sum of the natural numbers can be said to be equal to -1/12, traditionally it clearly diverges by blowing up to infinity but for example if we take analytic continuation seriously then the Riemann-Zeta function "tells us" that the sum of the natural numbers is -1/12, you will see many people claim that the Reimann-Zeta function is the only reason people assign this sum the value of -1/12, this is not true, there are multiple ways to assign this sum a value of -1/12
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Nov 13 '20
[deleted]
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u/c3534l Nov 13 '20
Isn't that what the meme is about? Isn't the second picture a knock-off that's not quite the same?
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u/KingRaunak Nov 13 '20
Quite a nice explanation
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Nov 13 '20
There is no explanation about the series diverging, he just handles it like it converges and gets that result, it will be only more confusing for the person.
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u/BartAcaDiouka Nov 13 '20 edited Nov 13 '20
u/Chand_laBing provides a precise explanation to the sum.
But if you are not advanced enough to understand it, let's say that, to be true, it requires new definitions for sums and limits. These new definitions are compatible with the properties you know about these two things, but they are broader and allow some divergent series (in the older definition, the one you know) to be convergent.
Edit: Also do not watch Numberphile's useless video about this sum, watch Mathologer's far superior take, which provide an actual explanation for it.
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u/aristotle2600 Nov 13 '20
Lot's of explanations for the math, but I think the joke itself needs a bit of elaboration. It's a play on the idea that a parent believes 2 things the same, when there are differences that the child making the request considers important. If you read the math explanations, the one thing you should take away is that the summation is not equal to -1/12 in the "normal way," but in a strange and new way.
Now compare this to a kid who wants a new toy, already possessing one that is similar. The child will be outraged and dismayed at the differences, while the parent will dismiss those differences as trivial. Who is right depends on context, just as it does with the math.
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u/Chand_laBing Nov 13 '20
As other comments have pointed out, this image macro is about the summation of divergent series and applying a summability method (e.g., Zeta analytic continuation) that assigns a value to a sum that would otherwise have been undefined. It essentially uses one of various possible redefinitions of how addition could work.
For example, you might accept that a geometric series can be summed as 1+x+x2+x3+…xn+… = 1/(1–x) as long as xn gets smaller as the sequence progresses, i.e., provided |x| < 1. If |x| < 1 weren't the case, e.g., choosing x = 2, then the series' terms would grow bigger: 1+2+4+8+… and the sum would seemingly equal infinity and would not converge. However, the expression on the right-hand-side would still make sense for x = 2 and would equal –1 there. So in this sense, we can say that if the relationship 1+x+x2+x3+…xn+… = 1/(1–x) did not just hold for small |x| but also held for large x = 2, then we would have the surprising equation 1+2+4+8+… = –1. In other words, we can extend the process of addition in such a way that values would be assigned in this way to those series and we are allowed paradoxical equations which have positive numbers summed to a negative.
The sum 1+2+3+4+… is essentially doing the same thing with other methods. I wrote a comment about this here, using what I find is the most convincing justification. Quoting myself,
So one way to justify the result to yourself would be that for all positive h, we have ∑_(1≤n≤N) ne–nh ≈ 1/h2–1/12. To then relate this to what happens at h=0, we can extend what we mean by summation and just consider that we would be left with the finite term.