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

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

#21  Postby archibald » Dec 01, 2016 10:16 am

newolder wrote:^ Perhaps the wiki on Cesaro summation will help?


After a quick scan of that, it seems that the summation to arrive at 0.5 is deemed 'illegal' because the series 1-1+1-1+1-1+1-1+1........ is neither convergent or divergent.
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Re: 1+2+3+4+5.... = -1/12

#22  Postby newolder » Dec 01, 2016 10:18 am

archibald wrote:
newolder wrote:^ Perhaps the wiki on Cesaro summation will help?


After a quick scan of that, it seems that the summation to arrive at 0.5 is deemed 'illegal' because the series 1-1+1-1+1-1+1-1+1........ is neither convergent or divergent.

Read just a bit more...
...
Therefore the Cesàro sum of the series G is 1/2.
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Re: 1+2+3+4+5.... = -1/12

#23  Postby archibald » Dec 01, 2016 10:23 am

newolder wrote:Read just a bit more...


But at the start it says..

(The Cesàro summation) "is commonly applied to Grandi's series (1-1+1-1+1-1+1-1+1........) with the conclusion that the sum of that series is 1/2, a result that can readily be disproven."
https://en.wikipedia.org/wiki/Ces%C3%A0ro_summation

It appears to be called a swindle..

"This "proof" is not valid as a claim about real numbers because Grandi's series 1 − 1 + 1 − 1 + ... does not converge"
https://en.wikipedia.org/wiki/Eilenberg ... ur_swindle
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Re: 1+2+3+4+5.... = -1/12

#24  Postby archibald » Dec 01, 2016 10:25 am

So...if you want to bring in averages, why not double the series then half it. You get 1, I think, without swindling. This answer also makes more sense.

ETA: No, that's bollocks. Sorry.
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Re: 1+2+3+4+5.... = -1/12

#25  Postby newolder » Dec 01, 2016 10:26 am

archibald wrote:
newolder wrote:Read just a bit more...


But at the start it says..

(The Cesàro summation) "is commonly applied to Grandi's series (1-1+1-1+1-1+1-1+1........) with the conclusion that the sum of that series is 1/2, a result that can readily be disproven."
https://en.wikipedia.org/wiki/Ces%C3%A0ro_summation

It appears to be called a swindle..

"This "proof" is not valid as a claim about real numbers because Grandi's series 1 − 1 + 1 − 1 + ... does not converge"
https://en.wikipedia.org/wiki/Eilenberg ... ur_swindle

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

#26  Postby archibald » Dec 01, 2016 10:30 am

newolder wrote:I know.


So, is it the case then that the problem is trying to get a number answer from something that isn't numerical (infinity)?
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Re: 1+2+3+4+5.... = -1/12

#27  Postby Galactor » Dec 01, 2016 10:38 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.


Or as mentioned in my post four posts earlier to your post. :mrgreen:
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Re: 1+2+3+4+5.... = -1/12

#28  Postby newolder » Dec 01, 2016 10:41 am

archibald wrote:
newolder wrote:I know.


So, is it the case then that the problem is trying to get a number answer from something that isn't numerical (infinity)?

I don't think so. The problem seems to be that G is divergent. The averages of the partial sums of G, however, tend to 1/2 and this is then defined as the Cesaro sum of G.
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Re: 1+2+3+4+5.... = -1/12

#29  Postby archibald » Dec 01, 2016 10:53 am

Cito di Pense wrote:No. Not correct. After the third fisherwoman leaves three...


I count that the third fisherwoman leaves 6?
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Re: 1+2+3+4+5.... = -1/12

#30  Postby archibald » Dec 01, 2016 10:59 am

newolder wrote:
archibald wrote:
newolder wrote:I know.


So, is it the case then that the problem is trying to get a number answer from something that isn't numerical (infinity)?

I don't think so. The problem seems to be that G is divergent. The averages of the partial sums of G, however, tend to 1/2 and this is then defined as the Cesaro sum of G.


But....we are not taking an average. I think it's a misuse of the word. We are instead arbitrarily stopping counting at two places and going halfway between those, which isn't the same thing as taking an average.
Last edited by archibald on Dec 01, 2016 11:21 am, edited 2 times in total.
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Re: 1+2+3+4+5.... = -1/12

#31  Postby newolder » Dec 01, 2016 11:15 am

archibald wrote:
newolder wrote:
archibald wrote:
newolder wrote:I know.


So, is it the case then that the problem is trying to get a number answer from something that isn't numerical (infinity)?

I don't think so. The problem seems to be that G is divergent. The averages of the partial sums of G, however, tend to 1/2 and this is then defined as the Cesaro sum of G.


But....we are not taking an average. I think it's a misuse of the word. We are instead arbitrarily stopping counting at two places and going halfway between those, which isn't the same thing as taking the average.

No. As shown at the wiki page, a new series is created from the partial sums that goes like:
1/1, 1/2, 2/3, 2/4, ... and the limit of this series is 1/2. This is then defined as the Cesaro sum of G.
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Re: 1+2+3+4+5.... = -1/12

#32  Postby archibald » Dec 01, 2016 11:19 am

newolder wrote:No. As shown at the wiki page, a new series is created from the partial sums that goes like:
1/1, 1/2, 2/3, 2/4, ... and the limit of this series is 1/2. This is then defined as the Cesaro sum of G.


Not following.

I didn't understand the 1st video, I don't speak french. But the second video starts with the summation of the G series (1-1+1-1+1-1+1...) as 0.5, which seems to me bogus, so the next steps don't matter.
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Re: 1+2+3+4+5.... = -1/12

#33  Postby newolder » Dec 01, 2016 11:26 am

archibald wrote:
newolder wrote:No. As shown at the wiki page, a new series is created from the partial sums that goes like:
1/1, 1/2, 2/3, 2/4, ... and the limit of this series is 1/2. This is then defined as the Cesaro sum of G.


Not following.

I didn't understand the 1st video, I don't speak french. But the second video starts with the summation of the G series (1-1+1-1+1-1+1...) as 0.5, which seems to me bogus, so the next steps don't matter.

The wiki page makes it clear. The 'swindle' is what you see so far. The solution is to take the limit of the series of partial sums means (oops!) and define that as the Cesaro sum of G. The series of partial sums means is called tn at the wiki. I cannot make it any clearer. Sorry.
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Re: 1+2+3+4+5.... = -1/12

#34  Postby archibald » Dec 01, 2016 11:56 am

newolder wrote:
archibald wrote:
newolder wrote:No. As shown at the wiki page, a new series is created from the partial sums that goes like:
1/1, 1/2, 2/3, 2/4, ... and the limit of this series is 1/2. This is then defined as the Cesaro sum of G.


Not following.

I didn't understand the 1st video, I don't speak french. But the second video starts with the summation of the G series (1-1+1-1+1-1+1...) as 0.5, which seems to me bogus, so the next steps don't matter.

The wiki page makes it clear. The 'swindle' is what you see so far. The solution is to take the limit of the series of partial sums means (oops!) and define that as the Cesaro sum of G. The series of partial sums means is called tn at the wiki. I cannot make it any clearer. Sorry.


So...the limit of all the means of the partial sums of the series tends towards 0.5 as we approach infinity. Is that it?
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Re: 1+2+3+4+5.... = -1/12

#35  Postby Cito di Pense » Dec 01, 2016 12:15 pm

archibald wrote:
Cito di Pense wrote:No. Not correct. After the third fisherwoman leaves three...


I count that the third fisherwoman leaves 6?


Is that relevant to the problem that the third fisherwoman leaves a different general situation than the other two?

This is called 'breaking symmetry'.

archibald wrote:
So...the limit of all the means of the partial sums of the series tends towards 0.5 as we approach infinity. Is that it?


It's pretty explicitly the case, isn't it, after induction on the first 2k partial sums.
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Re: 1+2+3+4+5.... = -1/12

#36  Postby archibald » Dec 01, 2016 12:20 pm

Cito di Pense wrote:
archibald wrote:
Cito di Pense wrote:No. Not correct. After the third fisherwoman leaves three...


I count that the third fisherwoman leaves 6?


Is that relevant to the problem that the third fisherwoman leaves a different general situation than the other two?

This is called 'breaking symmetry'.


No, it's called 'don't worry, there are only 3 fisherwomen'.
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Re: 1+2+3+4+5.... = -1/12

#37  Postby Cito di Pense » Dec 01, 2016 12:24 pm

archibald wrote:
Cito di Pense wrote:
archibald wrote:
Cito di Pense wrote:No. Not correct. After the third fisherwoman leaves three...


I count that the third fisherwoman leaves 6?


Is that relevant to the problem that the third fisherwoman leaves a different general situation than the other two?

This is called 'breaking symmetry'.


No, it's called 'don't worry, there are only 3 fisherwomen'.


Then there are multiple easy solutions to find, and you can get busy finding the next one.
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Re: 1+2+3+4+5.... = -1/12

#38  Postby archibald » Dec 01, 2016 12:25 pm

Cito di Pense wrote:
archibald wrote:
So...the limit of all the means of the partial sums of the series tends towards 0.5 as we approach infinity. Is that it?


It's pretty explicitly the case, isn't it..


Well it took me a while to get to it.

Cito di Pense wrote:.....after induction on the first 2k partial sums.


I only did 10. :(
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Re: 1+2+3+4+5.... = -1/12

#39  Postby archibald » Dec 01, 2016 12:27 pm

Cito di Pense wrote:Then there are multiple easy solutions to find, and you can get busy finding the next one.

-2?

That would seem to cover any number of fisherwomen.
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Re: 1+2+3+4+5.... = -1/12

#40  Postby Cito di Pense » Dec 01, 2016 12:30 pm

archibald wrote:
Cito di Pense wrote:
archibald wrote:
So...the limit of all the means of the partial sums of the series tends towards 0.5 as we approach infinity. Is that it?


It's pretty explicitly the case, isn't it..


Well it took me a while to get to it.

Cito di Pense wrote:.....after induction on the first 2k partial sums.


I only did 10. :(


That's more than 2k, for small values of k.
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