1+2+3+4+5.... = -1/12

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1+2+3+4+5.... = -1/12

#1  Postby aban57 » Nov 30, 2016 1:22 pm

This equation may seem familiar for many of you, but I discovered it with a video on facebook lately. Sorry it's in French, but I couldn't find the same in English. The demonstration is written anyway, so everyone can see it without problem. Go to 1:15 for the demonstration.

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Re: 1+2+3+4+5.... = -1/12

#2  Postby newolder » Nov 30, 2016 1:38 pm

There’s also the Numberfile versions:
1

and 2

(and there may be more) but an educational version is from Mathlogger:
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Re: 1+2+3+4+5.... = -1/12

#3  Postby Galactor » Nov 30, 2016 4:22 pm

It's a bit of a groan this one. The infinite series of -1 +1 -1 +1 ... converging to 1/2 is deeply dissatisfying. Or whatever it was ...
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Re: 1+2+3+4+5.... = -1/12

#4  Postby newolder » Nov 30, 2016 4:56 pm

^ That groanful series is known as Cesaro convergent. This is when the averages of the partial sums in a divergent series tend to a constant - in this case 1/2. The explanation begins around the 19 minute mark in Mathlogger’s video above. Satisfaction not guaranteed. :popcorn:
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Re: 1+2+3+4+5.... = -1/12

#5  Postby crank » Nov 30, 2016 5:05 pm

I've watched a number of these videos, some multiple times, can understand the proofs, I think, and I'm still dumb&awestruck, perplexed, queasy, and left with the feeling that there is something profoundly important lurking in there but I'll never be able to understand it. The equality is actually useful, in a number of situations, I think the one they usually mention is analyzing the Casimir Effect. It's simply crazy, like someone saying Donald trump could be the next US president.
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Re: 1+2+3+4+5.... = -1/12

#6  Postby Galactor » Nov 30, 2016 5:56 pm

newolder wrote:^ That groanful series is known as Cesaro convergent. This is when the averages of the partial sums in a divergent series tend to a constant - in this case 1/2. The explanation begins around the 19 minute mark in Mathlogger’s video above. Satisfaction not guaranteed. :popcorn:



Do you happen to know if there are practical applications of the series where the convergent result bears out somehow?
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Re: 1+2+3+4+5.... = -1/12

#7  Postby newolder » Nov 30, 2016 6:07 pm

Galactor wrote:
newolder wrote:^ That groanful series is known as Cesaro convergent. This is when the averages of the partial sums in a divergent series tend to a constant - in this case 1/2. The explanation begins around the 19 minute mark in Mathlogger’s video above. Satisfaction not guaranteed. :popcorn:



Do you happen to know if there are practical applications of the series where the convergent result bears out somehow?

Taming divergent series seems to be Cesaro’s only function but this guy at Quora has a few words to add: https://www.quora.com/What-is-the-signi ... -summation

The quote from Abel is probably apt:
“The divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever."
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Re: 1+2+3+4+5.... = -1/12

#8  Postby TopCat » Nov 30, 2016 6:13 pm

It's gems like this that keep me here, amidst all the dross of the current affairs and politics threads, which I have still found no easy way of suppressing. My thanks to the OP.
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Re: 1+2+3+4+5.... = -1/12

#9  Postby aban57 » Nov 30, 2016 6:17 pm

TopCat wrote:It's gems like this that keep me here, amidst all the dross of the current affairs and politics threads, which I have still found no easy way of suppressing. My thanks to the OP.


We're all here because we need this kind of food for our brain :)
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Re: 1+2+3+4+5.... = -1/12

#10  Postby Galactor » Nov 30, 2016 6:30 pm

newolder wrote:
Galactor wrote:
newolder wrote:^ That groanful series is known as Cesaro convergent. This is when the averages of the partial sums in a divergent series tend to a constant - in this case 1/2. The explanation begins around the 19 minute mark in Mathlogger’s video above. Satisfaction not guaranteed. :popcorn:



Do you happen to know if there are practical applications of the series where the convergent result bears out somehow?

Taming divergent series seems to be Cesaro’s only function but this guy at Quora has a few words to add: https://www.quora.com/What-is-the-signi ... -summation

The quote from Abel is probably apt:
“The divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever."


There are of course many mathematical constructions that appear abstract and beyond real application. But you only have to look at imaginary numbers and see what the imaginary resistance is, say, of a capacitor or an inductor, (their impedance) which we can calculate via the imaginary number i before you realise that abstraction in mathematics can "collapse" somehow to reality. Imaginary numbers explain real results.

The -1/12 universe depends upon this sequence and convergence which, I find, so undermines the calculation, but it would be more palatable if there were similar applications such as impedance from imaginary numbers.

As to where we go with the "understanding" that the answer to the universe and everything is not 42 but -1/12, you tell me.
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Re: 1+2+3+4+5.... = -1/12

#11  Postby newolder » Nov 30, 2016 6:53 pm

Galactor wrote:
newolder wrote:
Galactor wrote:
newolder wrote:^ That groanful series is known as Cesaro convergent. This is when the averages of the partial sums in a divergent series tend to a constant - in this case 1/2. The explanation begins around the 19 minute mark in Mathlogger’s video above. Satisfaction not guaranteed. :popcorn:



Do you happen to know if there are practical applications of the series where the convergent result bears out somehow?

Taming divergent series seems to be Cesaro’s only function but this guy at Quora has a few words to add: https://www.quora.com/What-is-the-signi ... -summation

The quote from Abel is probably apt:
“The divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever."


There are of course many mathematical constructions that appear abstract and beyond real application. But you only have to look at imaginary numbers and see what the imaginary resistance is, say, of a capacitor or an inductor, (their impedance) which we can calculate via the imaginary number i before you realise that abstraction in mathematics can "collapse" somehow to reality. Imaginary numbers explain real results.

The -1/12 universe depends upon this sequence and convergence which, I find, so undermines the calculation, but it would be more palatable if there were similar applications such as impedance from imaginary numbers.

As to where we go with the "understanding" that the answer to the universe and everything is not 42 but -1/12, you tell me.

The -1/12th bit arises in the analytic continuation of the Reimann Zeta function, as far as I can tell from the Mathlogger tube. So, our improved understanding of these series will impact our knowledge of the prime numbers and may well earn the "understander" a million bucks from the Millennium Prize Awardsters, iirc. Is that the kind of practical application you seek?
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Re: 1+2+3+4+5.... = -1/12

#12  Postby twistor59 » Nov 30, 2016 8:35 pm

We did this here in the pre LaTeX days before LateX came and went away again
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Re: 1+2+3+4+5.... = -1/12

#13  Postby newolder » Nov 30, 2016 9:37 pm

twistor59 wrote:We did this here in the pre LaTeX days before LateX came and went away again

Yes, completely forgotten about those posts. :doh: The equations look (mostly) ok in that topic. Is LaTeX working again since the recent server move? :dunno: I have e + 1 experience with LaTeX.
Last edited by newolder on Dec 01, 2016 12:20 am, edited 2 times in total.
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Re: 1+2+3+4+5.... = -1/12

#14  Postby tuco » Nov 30, 2016 10:34 pm

Galactor wrote:It's a bit of a groan this one. The infinite series of -1 +1 -1 +1 ... converging to 1/2 is deeply dissatisfying. Or whatever it was ...


Indeed, about as satisfying as square root of negative one :) as mentioned in one of the vids or perhaps even the three fisherwomen.
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Re: 1+2+3+4+5.... = -1/12

#15  Postby Cito di Pense » Dec 01, 2016 6:03 am

tuco wrote:
Galactor wrote:It's a bit of a groan this one. The infinite series of -1 +1 -1 +1 ... converging to 1/2 is deeply dissatisfying. Or whatever it was ...


Indeed, about as satisfying as square root of negative one :) as mentioned in one of the vids or perhaps even the three fisherwomen.


I'll write a python program. Beatrice's original solution assumes the third fisherwoman is the third one, but why should we assume that? It's a much harder problem if it doesn't matter where in the sequence the divvying up happens, because in any event there should be left a multiple of three, plus one, so I don't accept that I've seen a solution, yet.

tuco wrote:Correct :) By demonstration I meant how to arrive to solution, but "fiddle around with numbers" is sufficient.

Any other solutions, anyone?


No. Not correct. After the third fisherwoman leaves three, not assuming anyone else has come along, no one's going to be able to take one, divide by three and leave no one the wiser. Either this is not a solution, or every solution has to end up with a muliple of three at the end, in which case it's possible to come up with multiple solutions. But that isn't specified in the problem statement. So far, the person who thought this one up is not bucking for a job as a game developer.
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Re: 1+2+3+4+5.... = -1/12

#16  Postby tuco » Dec 01, 2016 7:17 am

Nerd joke newolder tsk tsk ;)

Wrong thread Cito di Pense. I have never said that I think you are not smart. However, multiple times I have said you waste your talent, especially in debates with those who probably are not very smart. The parallel I used, irrc, was: an adult playing football against a kid.
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Re: 1+2+3+4+5.... = -1/12

#17  Postby Cito di Pense » Dec 01, 2016 8:05 am

newolder wrote:
twistor59 wrote:We did this here in the pre LaTeX days before LateX came and went away again

Yes, completely forgotten about those posts. :doh: The equations look (mostly) ok in that topic. Is LaTeX working again since the recent server move? :dunno: I have e + 1 experience with LaTeX.


I luuuuuves me some LaTeX. I'd have way more fun here if it worked.
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Re: 1+2+3+4+5.... = -1/12

#18  Postby newolder » Dec 01, 2016 9:16 am

Cito di Pense wrote:
newolder wrote:
twistor59 wrote:We did this here in the pre LaTeX days before LateX came and went away again

Yes, completely forgotten about those posts. :doh: The equations look (mostly) ok in that topic. Is LaTeX working again since the recent server move? :dunno: I have e + 1 experience with LaTeX.


I luuuuuves me some LaTeX. I'd have way more fun here if it worked.

I've just checked a few of the posts with equations in that topic and, sad to report, they are not posted as LaTeX but use some of the extended, unicode characters. Heigh ho.
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Re: 1+2+3+4+5.... = -1/12

#19  Postby archibald » Dec 01, 2016 9:51 am

Amateur mathematician's tuppenceworth:

If, to get 1-1+1-1+1-1+1............to add to 0.5, we take the average of two possible answers (1 and 0) to a finite approximation of the total of an infinite series, isn't this inherently a botch, mixing two different things in the same calculation? How can the average between two finite sums tell us anything about an infinite one?

As for practical applications of such conundrums, the one that always puzzles me is if you have a square of side length x then the area of the square is x2, but this means that mathematically the sides can also be -x in length, but that's discarded as an impossible solution. Doesn't that suggest that there's something wrong with maths?
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Re: 1+2+3+4+5.... = -1/12

#20  Postby newolder » Dec 01, 2016 10:05 am

^ Perhaps the wiki on Cesaro summation will help?
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