I also see now that jtgain was thinking the proposal was that 1 + 2 + 3 + 4 + … equal 1/12. But the proposal is actually that, in a certain natural sense, this summation comes out to negative 1/12. (Yes, I know, that’s even more wild from a traditional perspective on summation. And certainly there are senses in which it is no such thing; in which the appropriate thing to say is that this sums to a positively infinite magnitude. But, as noted, there are also senses in which it can be appropriate to use the manipulations which assign this sum the value -1/12. There is no conflict; mathematics allows us to study both of these perspectives, and many more besides.)
I thought its accepted that you should show that the sum was ANYTHING using similar techniques.
On one hand, I was happy to finally find the way of shifting that would get me -1/12 instead of all the other answers I’d gotten that didn’t hold up. On the other hand, I was disappointed that they didn’t bother explaining how this was a completely different way to handle summation, and that the answer that the series converges at infinity is still correct. The second, more complicated video does better, but it’s still lacking.
I do wonder, Indistingishable, if all the people in the comments are right and that the method in the first video can get any answer. Well, I know they are right, because I sat up trying to figure it out in bed and got different answers. But are they right in that this shifting doesn’t actually prove anything?
It seems to me that the second video does prove it, but I’m starting to wonder if the first video is just comes from knowing the intended answer ahead of time and making it happen. Can you explain why this would not be so?
As for why people have trouble with this? Honestly, I just think the convergence method is intuitive and this isn’t. The convergence method just involves imagining pieces and then arbitrarily declaring an infinite number of them. It’s just easier to grasp.
I honestly can’t think of any conceptual way to make this math make sense. And, in real life, that is usually a sign that you’ve made a mistake. For example, if I do the math for the volume of my cup in front of me, and I get 500 gallons, I know I’ve made a mistake, as there’s no way for me to conceive my cup as holding that much. Or if I get an answer in square inches, which is not a measure of volume, I can’t conceive that, so I know it’s a mistake.
This sort of thing is why I thought of imaginary numbers as just being a parlor trick that happened to work in certain mathematical situations until someone (on this board, actually) gave me a way to conceptualize them.
Until you can come up with a way to for people to conceptualize the sum of all natural numbers to be -1/12, a vast majority are not going to accept it, even if the math keeps on working in multiple situations.
Re: “You can use different techniques to get different results”, yes, that is true. You can (for example, you can get positive infinity…). But there is a particular natural, systematic account of summation which blesses these manipulations and this answer, and not other manipulations leading to other answers. That was the purpose of my posts from #159 onwards in the old thread.
Specifically, we can look at it this way: We can try to assign values to a non-absolutely convergent series by bringing its terms in at less than full strength, producing an absolutely convergent series, and then increasing the terms’ strengths towards full strength in the limit, observing what happens to the sum in the limit as well.
This is the idea behind the traditional account of summation, mind you: at time T, we bring in all the terms of index < T at 100% strength and all other terms at 0% strength. This gives us our partial sums, and as T goes to infinity, each term’s strength goes to 100%, so we can consider the partial sums as approximating the overall sum.
But we don’t have to be so discrete as to only use 100% strength and 0% strength. We can try bringing in terms more gradually. This gives us the idea of Abel summation: rather than having strengths discretely decay from 100% to 0% at some cut-off point, we instead have the strengths decay exponentially in the index. (So at one moment, we may have the first term at 100% strength, the second term at 50% strength, the third term at 25% strength, etc.). Then we consider what happens as the rate of exponential decay slows, approaching no decay at all.
In symbols, this means we assign to a series a[sub]0[/sub] + a[sub]1[/sub] + a[sub]2[/sub]… the limit, as b approaches 1 from below, of a[sub]0[/sub]b[sup]0[/sup] + a[sub]1[/sub]b[sup]1[/sup] + a[sub]2[/sub]b[sup]2[/sup] + … Put another way, the limit, as h goes to 0 from above, of a[sub]0[/sub]e[sup]-0h[/sup] + a[sub]1[/sub]e[sup]-1h[/sup] + a[sub]2[/sub]e[sup]-2h[/sup] + …, where e is any fixed base you like. (Let’s call this function of h the characteristic function of the series).
Again, this is not so different than the traditional account of summation; we’re just using exponential decay rather than sharp cutoff in our dampened approximations to the full series.
Now we’ve turned the question of determining the value of a series summation into the question of determining the limiting behavior of some function at 0.
Well, it’s easy to determine limiting behavior at 0. Just write out a Taylor series centered at 0, and drop all the terms of positive degree, leaving only the term of degree 0. Boom, you’ve got the value of the function at 0.
Except… suppose the Taylor series has a few terms of negative degree as well. (As in, say, 5h[sup]-1[/sup] + 3 + 4h[sup]2[/sup]). Then the behavior at 0 isn’t given by the degree 0 term; rather, the behavior at 0 is to blow up to infinity!
And, indeed, we’ll find that this is precisely what happens we look at the characteristic function of a series like 0 + 1 + 2 + 3 + …; we get that f(h) = 0e[sup]-0h[/sup] + 1e[sup]-1h[/sup] + 2e[sup]-2h[/sup] + 3e[sup]-3h[/sup] + … = e[sup]-h[/sup]/(1 - e[sup]-h[/sup])[sup]2[/sup] = h[sup]-2[/sup] - 1/12 + h[sup]2[/sup]/240 - h[sup]4[/sup]/6048 + …
Note that there is a negative degree term there. So in a very familiar sense, we can say that the behavior of this series is to blow up to infinity.
However, since any time a series DOES converge in the ordinary sense, the value it converges to is the degree 0 term of this characteristic function, it is very tempting and fruitful to think of the degree 0 term as the sum even when there are those pesky negative degree terms.
And in this more general sense, we see that the value of 0 + 1 + 2 + 3 + … is that degree 0 term of f(h): -1/12.
Yes, you can do other shifts and manipulations to produce other answers in other ways, but this is one particular systematic account of summation which leads to this value alone and no other. I explain in the other thread why, on this account of summation, none of the other shifts and manipulations are justified.
Why should you care about this particular account of summation? Well, you don’t have to; I can’t force you to care about anything. But it’s fairly natural and comes up with some significance in mathematics. It is, in a sense spelt out in the other thread, precisely the account of summation which allows one to interpret the sum 1[sup]n[/sup] + 2[sup]n[/sup] + 3[sup]n[/sup] + … for arbitrary n, giving the general Riemann zeta function (of great significance in number theory, and whose behavior (specifically, the Riemann hypothesis concerning its zeros) is generally considered the most important open problem in mathematics).
So, yes, you can do other manipulations and get other values for 1 + 2 + 3 + 4+ … Absolutely, we should all acknowledge this. But these particular manipulations tie into a general, systematic theory which people care a lot about, in a way which those other ad hoc manipulations won’t.
Perhaps it might be easier to accept a simpler sum like this? Suppose we take 1 + 2 + 4 + 8 + 16 + … . In binary, this would be expressed as …111111111. Would it surprise you to learn that this is equal to -1? But it’s easy enough to see: Just add 1 to it and see what you get. In binary, 1+1 = 0, and you carry a 1 over to the next place. Where you again get 1+1, and another carry, and so on. So when you add 1 to that number, you get something whose digits are all zero, and whaddaya know, we recognize that number!
And in actual fact, this is actually the way a lot of computers represent negative numbers, and precisely for the reason that the rules for addition are nice and simple that way.
jtgain, I advocate going back and reading my post in this thread. As I said, in any sense that you learn about addition in elementary school, 1 + 2 + 3 + 4 + 5 + . . . has no answer at all. You aren’t told any way to add an infinite number of things at that point. Later, perhaps in high school or college, you’re told that it adds up to ∞. But what does that even mean? It’s not one of the natural numbers. Unless you’re taught a bunch of new rules to handle arithmetic with infinities, that’s a meaningless answer.
Essentially, every time you learn about new sorts of numbers, you learn ways to solve equations that you were told before were impossible to solve. So at first you’re told that you can’t subtract 4 from 3, but then they tell you about negative numbers, so the answer is -1. At first you’re told that if you divide 8 by 3 the answer is 2 with a remainder of 2, but then they tell you about fractions, so the answer is 2 2/3. At first they tell you that 3,468 divided by 100 is 34 17/25, but then they tell you about decimals, so the answer is 34.68. At first they tell you that x^2 = -4 has no answer, but then they tell you about complex numbers, so the answer is 2i.
So what we’re doing here is telling you that there’s a new kind of arithmetic which we will define. The following sums will have answers in this new arithmetic:
1 - 1 + 1 - 1 + 1 + . . .
1 - 2 + 3 - 4 + 5 + . . .
1 + 2 + 3 + 4 + 5 + . . .
In the old arithmetic that you learned before, the first sum has no answer, since the sum bounces back and forth between 1 and 0 as you go further and further in the sum. In the old arithmetic, the second sum has no answer, since the sum bounces back and forth from 1 to -1 to 2 to -2 to 3 to -3 to etc. In the old arithmetic, the third sum has no answer, since it increases without bound. (That is, unless you want to allow an answer of ∞, but we don’t want to allow such answers.) So what we’re telling you now is that we have a new kind of arithmetic. Each of those three sums has an answer now. The answers to finite sums aren’t different. The answers to infinite sums that converge aren’t different. Only the answers to infinite sums that don’t converge is different.
So, you ask, why do we want to create this new arithmetic? It’s because it solves problems in physics that otherwise we can’t solve. It’s not useful for anything you do, I assume. It is useful for physicists though.
In case anyone will call me out on this: I had earlier in this post taken e to be “any fixed base you like”, but here I’ve specifically taken e to be the base of the natural logarithm. It doesn’t really make any difference for anything; other bases will just scale h by a constant factor, keeping the degree 0 term unaffected.
You’re probably assuming (and correct me if I’m wrong here; despite what you said, I really don’t want to treat you like someone who’s 12 or brain-damaged) that the sum 1 + 2 + 3 + … is a simply defined thing, and a real number at that. That’s not the case. Suppose for a moment that you know nothing about infinite sums, and you’re trying to define one: that is, given a infinite sequence, you want to define its sum. One thing to note first is that mathematicians are (as you may have seen above) not really into definining the sum of a sequence. We consider them as maps from sequences into numbers that “look like” sums in a way I’ll clarify below, and we’re not interested in the philosophical issue of which is the One True Sum of the sequence; we’re interested in what happens when you apply one of these sum functions to a sequence [note 1].
So let’s try to define the sum of a sequence. The first problem is that although the sum of any two individual real numbers is well-defined, there isn’t an inherent notion of the sum of an infinite number of…numbers [note 2]. Induction gives you the sum of any finite number of reals, but there’s no inherent definition for an infinite sum. The obvious thing to try is the limit of the partial sums: for a sequence (a_1, a_2, …), but P_n = a_1 + … + a_n, and define the sum P(a_n) of the a_n to be limit of the P_n. This is not a bad definition; right off, we can see that P preserves positivity (i.e., if all the a_i >= 0, then P >= 0) and linearity (i.e., P(t a_n) = tP(a_n) for a constant t, and P(a_n + b_n) = P(a_n) + P(b_n). The rather glaring problem is that P may not exist. Not only can P diverge to infinity, but it may just fail to converge even if all the P_n are bounded. For example, take the famous sequence 1, -1, 1, -1,…
At this point, you may be tempted to call it a day and declare, “Well, if P doesn’t exist, then the sum isn’t defined, and it sucks to be you, anonymous mathematician who’s trying sum an infinite sequence.” This isn’t the worst possible case if the P_n go to infinity (but more about that later), but it is disappointing. It’s not unreasonable to try define P so that the sequence 1, -1, 1, -1, … converges to some finite value. (Again, that just means that P({1, -1,1, -1, …0}) is well-defined and finite for this mysterious function P that maps sequences to reals. I’m not saying anything about the sequence {1, -1, -1, …} inherently.) One method of doing so is via Cesaro summation, which is outlined here. As an added bonus, it turns out that any convergent sequence (in the usual sense) also has a well-defined Cesaro sum, and in fact the two coincide. This is encouraging.
Why are we doing this at all? Well, the honest reason is that because we’re mathematicians in this scenario, and this sort of thing is what mathematicians do. The less honest but more defensible reason is that summation is trickier than you might naively expect. As was mentioned earlier, the naive sum (i.e., the limit of the partial sums in order) is not independent of the order of the terms; if the series is conditionally but not absolutely convergent, then in fact any arbitrary real number can be obtained as a sum of some permutation of the sequence. This is not encouraging. More importantly, there are instances where the naive sum is definitely the wrong summation function P to use.
For example, consider Fourier series. (Sorry, that’s probably too much for an actual 12-year-old, but I don’t have a better example. There’s a decent outline on wikipedia, though.) In the simplest possible case— say, a continuous function on the circle (equivalently, a periodic function), you might expect the Fourier series to converge everywhere to original function. That is, in fact, false. (In fact, if you eliminate the continuity restriction and consider L^1 functions, there exist such functions whose Fourier series diverge almost everywhere.) There is a result (Fejer’s theorem), though, that states that the Fourier series of such a continuous function is Cesaro summable to the original function. We’re actually getting something out of this abstract nonsense.
Cesaro summation is pretty reasonable, but in certain circumstances it’s useful to consider more complicated things. One such method is zeta function regularization, as was discussed earlier; in more generality, you can look at summing something like g(z) = \sum a_n f_n(z) for some functions f_n(z) with (say) f_n(0) = 1, then look at the limit as z -> 0. In some cases, g behaves nicely on a suitable region of z and can be extended [note 3] all the way to z = 0. Unfortunately, that means that we can lose some nice, expected properties like positivity along the way. It’s not that mathematicians are unaware of this, as kaltkalt seems to imply; it’s just something that happens as expected. Indeed, it would be remarkable if this method would also preserve some of the nicer properties of more well-behaved sums. But, for complicated reasons that require more math than a 12-year-old is probably familiar with, we can use this method and the function g to produce meaningful, consistent, and sound proofs— even some involving perfectly ordinary sums.
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If you’re familiar with it, consider the corresponding notion for integrals. There are different things called integrals due to Riemann, Lebesgue, Stieltjes, and so on. We consider them all as ‘integrals’ for two reasons: they have the properties we expect integrals to have (linearity, positivity, etc.), and they agree in the simplest cases (say, continuous functions on a compact space).
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This may seem overly pedantic, but it really isn’t. If I want to define a infinite sum of rational numbers, I generally can’t without expanding it into real numbers; even the naive idea of summation doesn’t give me a rational number even I start with rational numbers. More damningly, suppose I’m working with something like Z/pZ (the integers mod p) or Z (the space of polynomials in a single variable with integral coefficients). I can add together pairs of elements in either of those rings perfectly well, but there isn’t a clear notion of what an infinite sum should be. (One fairly simple step is to look at the completion of a metric space, ring, etc. and compute the sum there; but that still leaves a couple of issues unresolved, and it’s going a bit outside the scope of this already overly-long post.
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I’m trying to avoid mentioning analytic continuation here, but that’s what’s actually going on. I can go into the details of why this method gives reasonable results and why this sort of extension is justifiable and meaningful if you’d like.
I’m the type of person who needs to see things worked out. You say it’s easy enough to see that adding 1 to your infinite series and see that it equals 0. But I do not find that easy to see. I can see it in the finite case where we are given a byte-limit, but I have trouble extrapolating that to a general idea that would continue working piece by piece. The second I extend the idea to an even higher byte limit, the math I did before becomes invalid.
For example, at 8 -bits per bytes I get 255 + 1 = 0, meaning 255 = -1. But the second I add another bit, I get 255 + 1 = 256, and it takes adding another 255 to get to 0, meaning 256 = -255. I don’t see how I can generalize the process. This is not like taking the limits of an infinite summation. I can easily start from where I left off and see how it still is getting closer and closer to the limit. That allows me to generalize it piece by piece, thus allowing the leap that having infinite pieces would still have the same result.
To put it another way, with limits and convergence, I can see that the gap between the limit and the actual number is constantly getting smaller, and I can generalize that, if it ever got to infinite, that gap would be infinitely small and thus non existent. And I seem close to being able to do this with what you are saying, but actually showing the work would help me so much.
It always does–that’s why I understood both Numberphile videos better than the previous thread. The thread had too much theory and not enough working out the steps.
This may be a really dumb question, but do you get the same result (-1/12) if you add 2+4+6+8+10+12…to infinity? Or 5+10+15+20+25…? Or 42+84+126+168…?
No, the same technique would give 2 + 4 + 6 + 8 + 10 + 12 + … = -2/12 (as it’s 2 * (1 + 2 +3 + 4 + …)), and 5 + 10 + 15 + 20 + 25 + … = -5/12, and 42 + 84 + … = -42/12, and so on.
You can, if you like, understand the 1 + 2 + 4 + 8 + … = -1 result as an instance of “the gap is constantly getting smaller” in a particular sense as well. It’s just that instead of using the normal account of size, we will consider the size of a nonzero rational number to be the power of 2 in its reciprocal (so, for example, the size of 8 would be 1/8, the size of 24 = 3 * 8 would also be 1/8, the size of 5/24 = 5/(3 * 8) would be 8, etc.), with the size of zero remaining zero. The distance between two numbers will be, as usual, the size of their difference.
On this account, we will find that the initial segment partial sums 1, 3, 7, 15, … have distance 1/2, 1/4, 1/8, 1/16, …, from -1. Thus, 1 + 2 + 4 + 8 + … = -1.
This may seem like nonsense, but it’s actually a very well-behaved notion of size and the start of a very fruitful branch of mathematics called p-adic analysis.
But everything is always relative to the interpretation of things one is interested in. This is one account of summation one can give. That does not delegitimize the account of summation in which 1 + 2 + 4 + 8 + … = positive infinity. And similarly for the titular series. And one can investigate many different senses of summation, and see which are clean, interesting, and/or useful in their various ways.
No because it is linear just like the summation of convergent series, so multiplying each term in the series by a constant will multiply the value of the series-summation method for that series by a constant
So, the Abel sum of those 3 series will be -1/6, -5/12 and -7/2 respectively.
Nitpick: the account of summation which gives 1 + 2 + 3 + 4 + … = -1/12 is not ordinary Abel summation, but a further augmentation on top of that.
Interesting, thanks.
Now what if each term increased by a larger and larger amount? For instance, what if it was 1+3+6+10+15+21+28+36…to infinity? Could that be figured out as well?
(Sorry if I’m asking dumb questions; I’m not a math guy or a physics guy, I’m just a guy.)
Except that the number we have here isn’t “The eight digits on the end are all 1”; it’s “all of the digits are 1”. If you extend it out another digit, it’s no longer 255 + 1 = 0; it’s 511 + 1 = 0.
In simpler terms: ordinarily, we say that a quantity is getting arbitrarily small if the position of its leftmost nonzero digit goes arbitrarily far to the right (so in some sense, all of its digits are stabilizing to zero, in left to right order). In p-adic analysis, we say that a quantity is getting arbitrarily small if the position of its rightmost nonzero digit goes arbitrarily far to the left (so in some sense, all of its digits are stabilizing to zero, in right to left order).
In the particular case of base 2, this will yield the result that 1 + 2 + 4 + 8 + … gets arbitrarily close to -1. (The partial sums are 1, 3, 7, 15, …, whose difference from -1 is 2, 4, 8, 16… . In base 2, this is 10, 100, 1000, 10000, … . Note how, in right to left order, each digit eventually becomes and forevermore stays zero).
More generally, in base p, this amounts to saying that p[sup]n[/sup] goes to 0 as n gets large. Which means the geometric series 1 + p + p[sup]2[/sup] + …, whose nth partial sum is (p[sup]n[/sup] - 1)/(p - 1), will go to -1/(p - 1) as n gets large.
Ok - I’m still having trouble understanding the video - I mean I understand the video, but - come on - just seems wrong :). But, I’ve learned long ago that if people that seem to know what they are doing say something is true (especially in math and other hard science areas) - it usually is.
Your explanation seems to make more intuitive sense. Before I go on trying to understand this more - can Indistinguishable (or someone else that seems to understand this) confirm whether or not Chronos’ binary example is correct and an appropriate analogy?
Chronos’s binary example is absolutely correct, on a particular account of what summation means, and an appropriate analogy in illustrating how we can give a natural, useful account of summation which might lead us to suppose a sum of positive values is negative.
The principle behind Chronos’s specific example is not exactly the same as the principle behind the 1 + 2 + 3 + 4 + … = -1/12 example, in that there are natural notions of summation which would validate Chronos’s result but not the titular one and vice versa (in a formal sense, Chronos’s example illustrates the possibility that the value of an increasing function can overflow from positive to negative after passing through a “pole”, while the titular example illustrates the possibility that the degree 0 term of an increasing function can turn from positive to negative precisely as the center lands on a “pole”), but there are also notions of summation which would validate both results. Chronos’s example is perhaps useful to see first because, for many, it will be simpler to grasp and seem closer to familiar mathematics than the arguments surrounding the titular result, though this is more a function of the way standard curricula are set up than any intrinsic property of the math.