## Re: I'm going to define a new operation and call it 6.

1

Treating a function as if it was a number is a really efficient way of making sure that your ability to do maths will never get beyond high school level. My subsconscious mind has often found it to be a good defence against understanding things which it hated or feared, but which for one reason or another my superego had decided it needed to learn.

Posted by: dsquared | Link to this comment | 01-20-14 4:26 PM
2

"I am not a number, I am the limit of an infinite series!"

Posted by: Ginger Yellow | Link to this comment | 01-20-14 4:34 PM
3

This explanation (linked from the one in the OP) is also good, as is this earlier one by the same author.

Posted by: teofilo | Link to this comment | 01-20-14 4:38 PM
4

So mathematicians are assholes?

Posted by: Moby Hick | Link to this comment | 01-20-14 5:14 PM
5

Consistent assholes.

Posted by: Kreskin | Link to this comment | 01-20-14 5:21 PM
6

4: Fundamentally.

Posted by: JP Stormcrow | Link to this comment | 01-20-14 5:54 PM
7

Aaaargh all the discourse around this is so annoying.

Posted by: essear | Link to this comment | 01-20-14 6:22 PM
8

I agree with 7, but probably for different reasons.

Posted by: Robert Halford | Link to this comment | 01-20-14 6:38 PM
9

Christ, what an infinite series.

Posted by: Natilo Paennim | Link to this comment | 01-20-14 6:42 PM
10

7: I thought the link in 3.1 was kind of neat and interesting. Is it also annoying?

Posted by: Sifu Tweety | Link to this comment | 01-20-14 6:52 PM
11

What did Richard Sherman say about math?

Posted by: fake accent | Link to this comment | 01-20-14 6:59 PM
12

I guess some of these are okay. The first thing I saw was Mark Chu-Carroll taking this as an excuse to bloviate about how stupid he thinks everyone else is, while completely overlooking the fact that there's some interesting math to explain here. The next couple things I saw were also basically saying "ha ha, all these stupid people think they can add positive numbers and get a negative number!"

Posted by: essear | Link to this comment | 01-20-14 7:01 PM
13

Aaaargh all the discourse around this is so annoying

Too early for new mouseover text?

Posted by: Kreskin | Link to this comment | 01-20-14 7:03 PM
14

Most of the divergent series I have to deal with converge faster than convergent series, so this one is a little weird.

Posted by: essear | Link to this comment | 01-20-14 7:09 PM
15

The post linked in the OP is fine too. I don't even know where the others that annoyed me are. People were linking them on FB.

Posted by: essear | Link to this comment | 01-20-14 7:11 PM
16

Is there a way to see mouseover text on a phone/tablet?

Posted by: x.trapnel | Link to this comment | 01-20-14 7:40 PM
17

On my iphone I held my finger over the image and the text came up, along with save options. That didn't users to happen.

Posted by: Eggplant | Link to this comment | 01-20-14 7:44 PM
18

In Chrome you can prepend

view-source:
to any URL and then look for the alt text.

view-source:http://www.unfogged.com/

Posted by: fake accent | Link to this comment | 01-20-14 8:31 PM
19

14: Divergent series that converge faster than convergent ones? Do you mean quickly converging to a non-trivial set of sub sequential limits? (Not trying to troll, genuinely confused and curious.)

The first thing I saw was the Mark Chu-Carroll thing, too. He was being too much of an ass to Phil Plait, even though Plait should have taken off his eyes-wide-and-full-of-wonder glasses a bit sooner.

Posted by: dalriata | Link to this comment | 01-20-14 8:41 PM
20

I last looked at maths when I was taking my second semester calculus exam as a first year undergrad, so forgive me if this sounds stupid. But couldn't it be said that -1/12 is the point at which the infinite series becomes infinite, since it's the sum of the infinite series' limit? That is, it's the sum of the convergent series which maps most closely to the infinite series but is not infinite. It could be viewed as the series' tipping point. The series in question becomes infinite at its inception. The area in which we can deal with it is extremely small -- the sum of the equation's limit is less than zero. To pretend that this means that the sum of the equation is less than zero is stupid, but knowing its limit is helpful in at least one application to physics.

Posted by: text | Link to this comment | 01-20-14 9:34 PM
21

I am going to add the popularity of this video to the monotonically increasing list of Boing Boing's crimes.

Posted by: snarkout | Link to this comment | 01-20-14 9:53 PM
22

19.1: It's called Carrier's Rule, and it's kind of a joke but not really. When you have an asymptotic series, often you get a really good approximation with a small number of terms even though the series diverges. (E.g. if your asymptotic series is a power series in x, but your function has an essentially singularity like exp(-1/x^2) at x -> 0, some number of terms in the power series can still get really close to the right answer.)

Posted by: essear | Link to this comment | 01-20-14 10:15 PM
23

But couldn't it be said

The words can be pronounced, yes.

Posted by: essear | Link to this comment | 01-20-14 10:16 PM
24

That is well explained. However, this explanation is very post-1800. If you look at how Euler wrote about this kind of thing it is argued much more like the "magic trick" version.

Posted by: Unfoggetarian: "Pause endlessly, then go in" (9) | Link to this comment | 01-20-14 11:21 PM
25

In college we used to have all sorts of fun coming up with strange misapplications of the zeta function. Once even confused our professor for a while. I really need to take some refresher courses.

Posted by: Grumbles | Link to this comment | 01-21-14 12:09 AM
26

20: I am confused - what do you mean by the sum of the limit?

Posted by: Benquo | Link to this comment | 01-21-14 7:05 AM
27

The words can be pronounced, yes.

I suppose I should say: isn't that actually what it means to take the sum of the limit of an infinite series? Isn't it incredibly stupid to pretend that the sum of the limit of the infinite series is the sum of the infinite series?

Posted by: text | Link to this comment | 01-21-14 7:06 AM
28

I suppose you could call it a magic trick, but you could also call it intellectual dishonesty, if you're actually pretending to be serious.

Posted by: text | Link to this comment | 01-21-14 7:16 AM
29

The standard definition as a limit of an infinite series really is a perfectly sensible definition of the infinite sum. To go back to the analogy that Plait muffed, if you add up a bunch of terms on your calculator, you'll get close to the limit.

Posted by: Walt Someguy | Link to this comment | 01-21-14 7:32 AM
30

So you can either be an asshole, or you can try to derive some meaning from the analysis, which is that the sum of the convergent series is the point at which the infinite series can no longer be dealt with.

Posted by: text | Link to this comment | 01-21-14 7:53 AM
31

None of the three series in the video are convergent. None of the three series of partial sums you can construct from each series is convergent.

Posted by: Robert | Link to this comment | 01-21-14 8:02 AM
32

As you approach the limit, 1+2+3+...= 1.5+2.5+3.5+.... That's the reason mathematicians round to the nearest integer so often.

Posted by: Moby Hick | Link to this comment | 01-21-14 8:14 AM
33

Your mother is divergent. (Not directed at anyone in particular.)

Posted by: JP Stormcrow | Link to this comment | 01-21-14 8:22 AM
34

22: Ah! That's interesting and not behavior I knew about. That quip seems most intuitive to me in the case where the series is alternating.

It might be clearer to say that the standard definition of an infinite sum is the limit of a sequence of finite sums (when those sequences converge, i.e. the difference between them goes to zero).

The point about the techniques being similar to pre-1800 analysis is a good one, as a lot of stuff that e.g. the Bernoullis did wasn't always well justified; they often ended up being right (and would verify as such by some independent means) but it wasn't until Cauchy that we had a well-formalized rigorous calculus. My recollection of trying to read Newton's Principia was that a lot of it felt more like empirical, inductive examination of the space of mathematics than math as it's practiced today.

Posted by: dalriata | Link to this comment | 01-21-14 8:26 AM
35

Pistols at dawn tomorrow, Stormcrow. You asymptote.

Posted by: Awl | Link to this comment | 01-21-14 8:28 AM
36

My recollection of trying to read Newton's Principia was that a lot of it felt more like empirical, inductive examination of the space of mathematics than math as it's practiced today.

Whereas *I* found that that my knowledge of geometry was a sad farce.

Posted by: Turgid Jacobian | Link to this comment | 01-21-14 8:29 AM
37

Your knowledge of geometry is Epic Movie?

Posted by: Moby Hick | Link to this comment | 01-21-14 8:31 AM
38

35: Right you are. Tired of your hyperbole.

Posted by: JP Stormcrow | Link to this comment | 01-21-14 8:34 AM
39

36: There was that, too. I only got through the first fifty or so pages before I decided that there were more productive ways to feel stupid and frustrated. At least the part that I covered, the geometry didn't seem to be so much beyond what I knew but speaking a different language. The way he used it was very different from how it was taught to me in high school, especially when he'd basically take a limit over a set of geometric diagrams.

Posted by: dalriata | Link to this comment | 01-21-14 8:34 AM
40

31: I haven't watched the video, but I did read two of the explanations, which described the actual mechanics of the proof. Obtaining the value of -1/12 requires converting the infinite series to its limit, which is a convergent series.

Posted by: text | Link to this comment | 01-21-14 9:10 AM
41

I'm no mathematician, but no one else has chimed in, so I'll attempt to explain what I understood and maybe someone else will correct me.

Start with the formal series which is the sum of n^(-x) where n goes to infinity. We say formal series because this is not a series in itself. If you plug in a value of x which is greater than one, you'll get an infinite series which converges. If you plug in a value less than one, you'll get an infinite series which diverges, and as such does not have a limit.

Now, you can define a function with that formal series, but this function will only be well-defined for x greater than one. For every such x, the value of the function at x is simply the limit of a certain convergent series.

Finally, you can attempt to extend this function to x less than one in some way which is 'locally consistent'. This extension will not necessarily look like the original formal series for those other values of x, and in fact it doesn't. And then you can evaluate the extension at x=1, and obtain -1/12. But saying that this is "converting the infinite series to its limit" is inaccurate.

Posted by: Awl | Link to this comment | 01-21-14 9:58 AM
42

saying that this is "converting the infinite series to its limit" is inaccurate

In the specific example we're dealing with, the mathematicians are obtaining the sum of the convergent series which most approximates an infinite series. Then they are assigning that sum to the infinite series. If you don't like "conversion" don't use it, but I think it's fairly apt when you treat a series as if it were a series which is derived from it.

Posted by: text | Link to this comment | 01-21-14 10:08 AM
43

In the specific example we're dealing with, the mathematicians are obtaining the sum of the convergent series which most approximates an infinite series.

No, that's not what's happening. Awl's explanation in 41 is right.

Posted by: essear | Link to this comment | 01-21-14 10:23 AM
44

"converting the infinite series to its limit" is terminologically really confusing, and I didn't understand what 40 meant at all until I went back and read the link in 3. "its limit" implies that the limit is a property of the series, but a divergent series either doesn't have a limit or its limit exists only in the extended reals (according to the particulars of the situation and local custom). Rather, you're treating that divergent series as a limit of a particular sequence of convergent series, and then assigning to it (and, in the video, equating to it) the limit of the sums of those series (a value).

Posted by: dalriata | Link to this comment | 01-21-14 10:26 AM
45

43: sure, right, that.

Posted by: text | Link to this comment | 01-21-14 10:29 AM
46

41.last: It certainly doesn't look like the original formal series, but it is interesting that the value you get by this process is equal to the average of the subsequential limits. I wonder under what situations that holds true.

Posted by: dalriata | Link to this comment | 01-21-14 10:29 AM
47

I wonder under what situations that holds true.

Careful, you're about to turn this thread into an actual research project.

Posted by: essear | Link to this comment | 01-21-14 10:32 AM
48

So long as it's someone else's research project.

Posted by: dalriata | Link to this comment | 01-21-14 10:34 AM
49

Since this thread is still on topic, I'll note that it reminds me of this.

Posted by: Moby Hick | Link to this comment | 01-21-14 10:34 AM
50

Recently a few people in my approximate field have picked up on some kind of relatively new (by physics standards, anyway) branch of math about "resurgence" for asymptotic series and "transseries" and it somehow is a super-powerful generalization of Borel summation that they say makes rigorous a lot of the hand-wavy stuff physicists have been doing for decades. I keep intending to get around to learning about it one of these days, but I say that about everything.

Posted by: essear | Link to this comment | 01-21-14 10:35 AM
51

"converting the infinite series to its limit" is terminologically really confusing

If it confuses you, then I guess you're confused. The sum of the convergent series which approximates the infinite series in question for a finite set of numbers is -1/12. The term assigned to a convergent series which most approximates an infinite series for a finite set of numbers is its limit. The sum of the limit of the infinite series in question is -1/12. We might understand this to mean that the series cannot be dealt with, that it becomes infinite, at its inception.

Or we might not understand that at all. Take your pick.

Posted by: text | Link to this comment | 01-21-14 10:35 AM
52

somehow is a super-powerful generalization of Borel summation that they say makes rigorous

Good, because that shit bugged me.

Posted by: Turgid Jacobian | Link to this comment | 01-21-14 10:36 AM
53

49: That's a really useful xkcd deep cut. I should bookmark that.

51: What is the convergent series that sums to -1/12? Can you write out a few of its terms, or a formula for them? I'm admittedly more concerned with the one that goes to 1/2 earlier in the proof, if you think there's any such series. Your second sentence makes me think we're using words in interestingly different ways; do you mean to imply that a convergent series is not an infinite series? What's the importance of the "finite set of numbers" clarification?

Posted by: dalriata | Link to this comment | 01-21-14 10:45 AM
54

53: I was being overly simplistic. It's been a long time since I've looked at this stuff. The issue might be that the proof relies on taking the limit of an oscillating series in order to derive a value for the infinite series 1+2+3 . . . . In deriving the infinite series from the limits of the oscillating series, rather than the oscillating series themselves (which we can't do) we are essentially deriving the limit of the infinite series rather than the infinite series itself. So the maybe the convergent series can be said to be the equation which is derived from the limits of the oscillating series which lead us to the value of -1/12.

It would remain true, in that circumstance, that the value of -1/12 is the infinite series' limit -- it's the point after which we cannot deal with the infinite series. And it would make sense for that point to be less than zero, as the series in question moves rapidly to infinity.

Posted by: text | Link to this comment | 01-21-14 11:10 AM
55

I'm lead to this conclusion by the first link in 3, which explains things pretty well. I believe I've seen all this stuff before, but it was a long time ago, I want to say high school.

Posted by: text | Link to this comment | 01-21-14 11:14 AM
56

I've never seen this stuff in school, and haven't had the curiosity to do what it would take to really learn it. I find 41 to be well matched with my level of interest, and very nicely explained. Thanks, Awl.

Posted by: Eggplant | Link to this comment | 01-21-14 11:25 AM
57

The argument would go as follows: In deriving a single numerical value from an oscillating series, what you're really doing is taking its limit. If you then do algebra with the limits of the oscillating series to obtain a value for the infinite series, what you are really doing is obtaining the sum of the limit of the infinite series. This is all just a restatement of the explanation in the link in 3. The only thing that I'm adding is that the sum of the limit of the infinite series might be considered the point after which it is infinite, or the range of values in which we can actually deal with it. I want to say we did this in tenth grade.

Posted by: text | Link to this comment | 01-21-14 11:33 AM
58

No, that was 11th grade, sorry.

Posted by: text | Link to this comment | 01-21-14 11:34 AM
59

54: The issue (or at least my issue) is associating the limit of the sums of geometric series in the neighborhood (in some sense) of the alternating series 1-1+1... with that alternating series itself. The link at 3 also thinks that this is sketchy: "...which is the naive result claimed to be true in the video." (Admittedly, after that they get a bit hand-wavy--when they say they're summing an "infinitely close but distinct series" that means they're taking a limit in the space of series.) I think this is also Awl's issue above.

I still don't know what you mean by "the point after which it is infinite," which sounds entirely meaningless to me regardless of whatever terminological differences we have.

Posted by: dalriata | Link to this comment | 01-21-14 12:16 PM
60

59.last: yep

Posted by: Turgid Jacobian | Link to this comment | 01-21-14 12:25 PM
61

59 last: Well you could imagine a different, more complicated infinite series which initially increases more gradually, or even appears to approach some positive number, before its values increase rapidly. I think an equation can be written which would result in such a series, though I can't provide you with an example. Someone more active in math could probably provide one.

Anyway, if you wanted to define the area in which that series can be dealt with, essentially treated as if it were a convergent series, then you might use the same method that is being used here for 1 + 2 + 3 . . . . You might define a range of oscillating series which could be combined into the infinite series, take the limits of those oscillating series, and then solve for the missing value, through the magic of algebra.

But 1 + 2 + 3 . . . is a very basic series which doesn't converge before it reaches infinity. So if you are solving for a value at which it converges, you will get a value which basically tells you: before it starts. Here, that value is -1/12.

Posted by: text | Link to this comment | 01-21-14 12:30 PM
62

So yes, "the point after which it is infinite" is inapt. I should say, "the point after which we must treat it as infinite." And the point after which we must treat it as infinite is -1/12 because it doesn't converge on any value at any point before approaching infinity.

Posted by: text | Link to this comment | 01-21-14 12:31 PM
63

Crooked Timber has a thread on this that, in a reverse of the usual phenomenon, isn't nearly as bad as this one. At least the word salad there can almost be made edible with a little dressing.

Posted by: essear | Link to this comment | 01-21-14 12:32 PM
64

63: if you separate out the turds in this particular salad I'm sure it would be at least healthy and digestible.

Posted by: Sifu Tweety | Link to this comment | 01-21-14 12:35 PM
65

On 59 first, I think we are approaching comity. I agree that it is problematic to assign a value to the oscillating series which is simply an average of its two points of oscillating. In doing that, you're essentially taking the limit of the oscillating series. And you aren't going to be able to calculate the value of the original infinite series from the value of the limits of two oscillating series. But maybe what you can do, and are doing, is obtaining the limit of the infinite series. And its limit would be the range in which we can deal with it; i.e., the range in which we can treat it as if it weren't infinite.

Posted by: text | Link to this comment | 01-21-14 12:35 PM
66

63: yes, it reflects well to simply write "word salad," when you are wrong, or don't understand something, or both.

Posted by: text | Link to this comment | 01-21-14 12:36 PM
67

I agree that it is problematic to assign a value to the oscillating series which is simply an average of its two points of oscillation.

Posted by: text | Link to this comment | 01-21-14 12:38 PM
68

You tell 'em, text. With a JD and your memory of high school math, there's no problem you can't solve.

Law degrees. Is there anything they can't do?

Posted by: LizardBreath | Link to this comment | 01-21-14 12:39 PM
69

62: Why -1/12? Why not 0, or -1000, or negative infinity? You can't tell whether or not a series is going to converge by looking at finitely many terms. More generally, a sequence a_1, a_2, ... converges if for all epsilon > 0 there exists an N such that for all n,m > N, |a_n-a_m|

That "arbitrary small" is the kicker. And it needs to apply across the full domain (the natural numbers). There's no way that for-all up there will work if you're only analyzing a finite prefix.

Posted by: dalriata | Link to this comment | 01-21-14 12:42 PM
70

LB, I suspect you're not being serious with me all the time.

Posted by: text | Link to this comment | 01-21-14 12:43 PM
71

Why -1/12? Why not 0, or -1000, or negative infinity?

Because that's the value obtained when you solve for the original infinite series from the limits of two oscillating series which, in their original form, could be combined into the infinite series. That part's in the video.

Posted by: text | Link to this comment | 01-21-14 12:45 PM
72

You can't tell whether or not a series is going to converge by looking at finitely many terms.

I'll have to confess that this doesn't make a lot of sense to me. Generally it is possible to determine whether a series is convergent or not, while it remains impossible to look at an infinite number of terms.

Posted by: text | Link to this comment | 01-21-14 12:47 PM
73

The fuck?

Posted by: Moby Hick | Link to this comment | 01-21-14 12:50 PM
74

Law degrees. Is there anything they can't do?

Try chewing a law degree next time you're car sick.

Posted by: JP Stormcrow | Link to this comment | 01-21-14 12:56 PM
75

That's easy--it's no different than being able to make a claim about all numbers between 0 and 1, even though there's quite a few of them. As an example, onsider the series \sum_{i=0}^\infty 2^-i = 1+1/2+1/4..., which sums to 2. If we denote the partial sums 1 = S_0, 1+1/2 = S_1, 1+1/2+1/4 = S_2, etc, we see that

|S_m - S_n|
= |\sum_{i=n}^m 2^-i|
= |2^-n + ... + 2^-m|
= 2^-m * | 2^(m-n) + ... + 1|
= 2^-m * | 2^m-n+1 + 1|
= 2^-n+1 + 2^-m

(Probably some sign errors in there, but I'm in a hurry.)
We can bound that as close to zero as we like by picking a big enough n, so thus we can talk about all the terms, even though there's infinitely many.

Posted by: dalriata. | Link to this comment | 01-21-14 12:56 PM
76

75: I would quibble that you aren't looking at an infinite number of terms, you're finding means to express them in a finite language. And, while I am largely ignorant in this area, I would think that you would be able to express 1+2+3 . . . in finite language as well -- Y = n+1 or something like that -- without it being necessary to say that the entire sum is really -1/12.

Thanks for discussing this with me, given the limitations of my knowledge.

Posted by: text | Link to this comment | 01-21-14 1:01 PM
77

This needs an ear worm...and Muppets.

Posted by: md 20/400 | Link to this comment | 01-21-14 6:01 PM