Counter-intuitive Summation

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Re: Counter-intuitive Summation

#21  Postby Pulsar » Jan 15, 2014 9:54 pm

There's something quite pleasing about the fact that

S = 1 + 2 + 4 + 8 + 16 + ... = -1

Look at how S would be written in binary:

1 = ...000001
1 + 2 = ...000011
1+ 2 + 4 = ...000111
...
S = ...111111

So S looks like an infinite string of '1s'. What would S + 1 be? The rightmost digit would become 0. But the digit next to it would also become 0, and so forth

So S + 1 = ...000000 = 0,

therefore S = -1.

So S+1 is some sort of binary overflow :)

You can do the same thing with decimals:

S = 9 + 90 + 900 + 9000 + ...
10*S + 9 = 9 + 10*(9 + 90 + 900 + 9000 + ...) = 9 + 90 + 900 + 9000 + ... = S
10*S + 9 = S
9*S = -9
S = -1

Or in decimals: ...999999 + 1 = ...000000 = 0 -> ...999999 = -1


Here's another weird consequence: consider

S1 = 1 - 1 + 1 - 1 + 1 ... = 1/2

What happens if we put a zero between every pair 1 - 1?

S2 = 1 + 0 - 1 + 1 + 0 - 1 ...

What is S2? Use the same trick as in the video: add S2, but shift the terms 1 spot to the right. Then add S2 again, but now shift the terms 2 spots:
S2 + S2 + S2 = 1 + 0 - 1 + 1 + 0 - 1 ...
S2 + S2 + S2 = 1+ 1 + 0 - 1 + 1 + 0 - 1 ...
S2 + S2 + S2 = 1 + 0+ 1 + 0 - 1 + 1 + 0 - 1 ...

You'll find that 3*S2 = 1 + 1 + 0 + 0 + 0 ... = 2.
Therefore, S2 = 2/3.
So simply adding a zero between every pair 1 - 1 changes the sum from 1/2 to 2/3 :)
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Re: Counter-intuitive Summation

#22  Postby VazScep » Jan 15, 2014 10:47 pm

Pulsar wrote:
VazScep wrote:There are principled reasons for allowing non-Euclidean geometries and permitting a square root of -1.

There are principled reasons for these series too, see the posts from Twistor and Cali.
I read them. Cali's post just outlines how to define the zeta function by analytic continuation without addressing the crucial question of what this has to do with defining 1 + 2 + 3 + 4 ....

I know what the zeta function is. I know how to define it by analytical extension to the complex plane. I took complex analysis. I see no reason to use it as a basis for defining the original infinite sum, which was my objection from the beginning. Near as I can tell, Cali ends up agreeing with me on this:

Calilasseia wrote:But of course, the analytically continued version of the Riemann zeta function departs from the original power series definition outside the region s>1, because the analytically continued function is built from pieces involving other power series. At least, that's my understanding of the issue.
My emphasis.

Until a reason has been given to define the values of infinite sums in this way, and until some theory is presented about such sums, I'm going to continue to call bullshit. The second video is no better than the first: just some misleading algebraic substitutions.
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Re: Counter-intuitive Summation

#23  Postby VazScep » Jan 16, 2014 12:13 am

Standard analysis is all about banishing the infinite from the conversation. Infinite sums aren't really infinite sums. The infinite symbol and the ellipsis which pretends to be an infinitely long expression, those are just relics that we've kept over from the less rigorous days because they're notationally convenient. Instead, us modern types say that when you talk in infinite sums, you're really talking about limits of partial sums, each of which is finite. When you write down the value of an infinite sum, you're not talking about the result of a hypertask. You're not literally performing infinitely many summations. You're just talking about limits. Calculus is no less beautiful when you realise that all talk of infinite sums is cashed out in these less exotic terms.

So what are we talking about when we ask for the value of 1 + 2 + 3 + 4 + .... It's not an infinite summation, because calculus has done away with that crap. But it's not even the limit of partial sums, because such a limit doesn't exist.

Since we're messing with complex numbers, here's a comparison: the opening definition of exponentiation, raising one number to the power of another, is going to start with the positive whole numbers, and say something quite simple like:

mn = m * m * m ... * m

where there are n copies of m on the right-hand side. Somehow, by the time we've got to the complex numbers, we've landed at the far less intuitive but no less correct definition:

zw = ew * ln(z)

To use this, we need to know both how to compute ez for complex z and how to compute ln(z). And for the former, someone's probably suggested

ez = ex(cos(y) + i*sin(y))

for z = x + iy.

Where the fuck does this come from? Euler hit on it, but I suspect that he did so just by squinting really hard at some power series, and that he didn't have much of an argument to back it up. The explanation of this I was given at A-level is just the same bullshit pulled in these videos. It's all just notational guesswork.

Modern mathematicians are better than that. :) I expect to know the principled story of why we allow this definition starting from our naive definition about the positive whole numbers. And it goes by first extending the naive definition to the rational numbers by demanding that exponentation satisfies certain algebraic laws, and then to the real numbers by demanding continuity over the top, and finally to the complex numbers by demanding that the functions are everywhere differentiable. Those abstract demands turn out to have only one solution, and it happens to agree with Euler's fiat definition.

The way Euler got to the definition, by squinting at power series, is all still good intuition, but it's weak as a justification. What you need is the abstract specification of the solution, the proof that it is unique, and then it doesn't matter what intuition you use to get you there. You've got the solution, now you can rest happy.

That story is missing for 1 + 2 + 3 + 4 + .... So far, I see a bunch of intuitions and squinting, but no theory.

People didn't sit down and decide to invent non-Euclidean geometry or complex analysis. Discoveries are made in more haphazard ways, by trying out things and seeing what happens. Motivations are filled in after the fact.
A bunch of guys working on solving cubic equations in the 16th century sat down and started messing around with imaginary numbers. The work was justifiably regarded as convenient bullshit until William Rowan Hamilton showed how to exhibit the complex numbers as pairs of reals, and is now even more justified when we think of complex numbers as the algebraic completion of the reals. Similarly, a bunch of guys in the 18th century sat down and starting messing around with geometry whilst denying the parallel postulate. Their work was justifiably regarded as mere bullshit until a model of non-euclidean geometry was exhibited, and folk like Riemann gave a general account of geometries.

Maybe there's something interesting going on here with these divergent series, but given the arguments I've read so far, we're at best at the incubator stage.
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Re: Counter-intuitive Summation

#24  Postby scott1328 » Jan 16, 2014 12:20 am

All I want to know is does this mean really and for true that .999... = 1?
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Re: Counter-intuitive Summation

#25  Postby twistor59 » Jan 19, 2014 8:34 am

OK, more from the physics point of view, but Lubos Motl, who is

Lubos Motl wrote:a [right-wing] as*hole


has just posted on this.
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At last we hear from a biologist!

#26  Postby scott1328 » Jan 19, 2014 4:42 pm

P.Z. Myers has weighed in on this matter.

Oh, good. There’s this claim going around that the sum of all natural numbers (1+2+3+4+5…) converges on the value -1/12. I saw that and said to myself that it’s obviously wrong, but saw the smooth patter and rapid-fire use of mathematical jargon and infinities, and no mathematician myself, couldn’t see where the error slithered in.


The first error slithered in when Myers misrepresents the numberphile as claiming that the series converges, when, in fact, the video repeats several times that the series diverges.

Maybe someone should ask a geologist what he/she thinks of the mathematics in this result.
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Re: Counter-intuitive Summation

#27  Postby Thommo » Jan 19, 2014 5:16 pm

I have to admit I have been quite engaged by this idea, the result is surprising and though I still disapprove of calling something a proof when it isn't a proof (in fact it looks like an abuse of notation really) there does appear to be some interesting mathematics here. I guess numberphile has done its job.

I agree with your comment there Scott, PZ Myers should not be criticising a claim that was never made (that the sequence converges), likewise the blog post he links to seems similarly guilty for complaining that the numberphile video makes claims of sequences having Cesaro sums that I do not recall them making. Numberphile should shoulder some of the burden here for claiming that the sequence "is equal to" -1/12 instead of defining an operation on sequences which maps that sequence to -1/12 and thus obscuring what really goes on, but clearly nothing will be made clearer by more people such as Myers piling in more errors.

I also wonder whether the inconsistency of approach between the simple declaration of the sum of S1 compared to the algebra involved in S2 doesn't contribute to the feeling of unease the video gives to some people. Clearly instead of the approach they used they could instead have assigned S1 a value by algebraic manipulation:-

S1 = 1 - 1 + 1 - 1 + 1 - ...
1 - S1 = 1 - (1 - 1 + 1 - 1 + 1 - ...) = 1 - 1 + 1 - 1 + 1 - ... = S1

Thus 2S1 = 1, S1 = 1/2
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Re: Counter-intuitive Summation

#28  Postby Pulsar » Jan 19, 2014 10:53 pm

Thommo wrote:Clearly instead of the approach they used they could instead have assigned S1 a value by algebraic manipulation:-

S1 = 1 - 1 + 1 - 1 + 1 - ...
1 - S1 = 1 - (1 - 1 + 1 - 1 + 1 - ...) = 1 - 1 + 1 - 1 + 1 - ... = S1

Thus 2S1 = 1, S1 = 1/2

They did that in the other Numberphile video that was mentioned in the clip:



As simple as the proof is, many people clearly don't understand infinite series. Many commenters on that video think they can disprove the result, and fail completely (despite getting massively upvoted; one comment has 91 votes as I write this). I've never seen a Dunning–Kruger effect on such a massive scale. Fascinating.
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Re: Counter-intuitive Summation

#29  Postby Pulsar » Jan 19, 2014 10:53 pm

twistor59 wrote:OK, more from the physics point of view, but Lubos Motl, who is

Lubos Motl wrote:a [right-wing] as*hole


has just posted on this.

Classic Motl rant :lol:
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Re: Counter-intuitive Summation

#30  Postby Thommo » Jan 20, 2014 12:01 am

Pulsar wrote:As simple as the proof is, many people clearly don't understand infinite series. Many commenters on that video think they can disprove the result, and fail completely (despite getting massively upvoted; one comment has 91 votes as I write this). I've never seen a Dunning–Kruger effect on such a massive scale. Fascinating.


Interesting. Thanks for sharing.
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Re: Counter-intuitive Summation

#31  Postby ughaibu » Jan 22, 2014 7:11 am

Pulsar wrote:There's something quite pleasing about the fact that

S = 1 + 2 + 4 + 8 + 16 + ... = -1

Look at how S would be written in binary:

1 = ...000001
1 + 2 = ...000011
1+ 2 + 4 = ...000111
...
S = ...111111

So S looks like an infinite string of '1s'. What would S + 1 be? The rightmost digit would become 0. But the digit next to it would also become 0, and so forth

So S + 1 = ...000000 = 0,

therefore S = -1.
This is very nice, but it seems to be about something other than integers and addition, I assume this is what VazScep was getting at. You need to say something about what your symbols and operations are, otherwise your result appears to be false by contradicting the well ordering of the integers.
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Re: Counter-intuitive Summation

#32  Postby Pulsar » Jan 22, 2014 7:34 am

That post was just play, it wasn't meant to mean anything rigorous. I'm well aware that something like ...111111 is abus de notation.

But the Cesàro sum and Abel sum of 1 + 0 − 1 + 1 + 0 − 1 + ... are indeed 2/3, as mentioned here.
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Re: Counter-intuitive Summation

#33  Postby Calilasseia » Jan 22, 2014 10:46 pm

Returning back to the series:

1+2+3+4+5+6+7+ ...

According to the definition of the Cesàro sum, said sum does not exist for this series, because the arithmetic means of the partial sums diverge too.
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Re: Counter-intuitive Summation

#34  Postby JSS » Jan 23, 2014 1:01 pm

Isn't it true that "all infinities are not alike"?

If you do anything to change the number of terms in an infinite series, you can still claim that it is infinite, but not identically infinite, not "equal to" the original.

[1+1+1+...+1] =/= [1+1+1+...+1] + 1
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Re: Counter-intuitive Summation

#35  Postby ughaibu » Jan 23, 2014 1:03 pm

JSS wrote:Isn't it true that "all infinities are not alike"?

If you do anything to change the number of terms in an infinite series, you can still claim that it is infinite, but not identically infinite, not "equal to" the original.

[1+1+1+...+1] =/= [1+1+1+...+1] + 1
Both those infinities are countable, so they're the same size.
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Re: Counter-intuitive Summation

#36  Postby JSS » Jan 23, 2014 1:06 pm

How do you measure "size" such that they are the same size?
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Re: Counter-intuitive Summation

#37  Postby ughaibu » Jan 23, 2014 1:10 pm

[url=http://legacy.earlham.edu/~peters/writing/infapp.htm]A Crash Course in the Mathematics
Of Infinite Sets[/url] I've no idea why the link fucked up: http://legacy.earlham.edu/~peters/writing/infapp.htm
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Re: Counter-intuitive Summation

#38  Postby ughaibu » Jan 23, 2014 1:12 pm

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Re: Counter-intuitive Summation

#39  Postby JSS » Jan 23, 2014 1:20 pm

So, you can't explain it?

It seems to me pretty obvious that if you add something to anything, you necessarily have something different, else you didn't really add anything.

Thus for n!=0, f{x}+n =/= f{x}
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Re: Counter-intuitive Summation

#40  Postby Matt_B » Jan 23, 2014 1:33 pm

JSS wrote:How do you measure "size" such that they are the same size?


http://en.wikipedia.org/wiki/Aleph_number

I prefer cardinality, as it's much less ambiguous.
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