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.
Moderators: Calilasseia, ADParker
...
Therefore the Cesàro sum of the series G is 1/2.
newolder wrote:Read just a bit more...
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
newolder wrote:I know.
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.
Cito di Pense wrote:No. Not correct. After the third fisherwoman leaves three...
newolder wrote:
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.
archibald wrote:newolder wrote:
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.
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.
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.
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.
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?
Cito di Pense wrote:
Is that relevant to the problem that the third fisherwoman leaves a different general situation than the other two?
This is called 'breaking symmetry'.
Cito di Pense wrote:.....after induction on the first 2k partial sums.
Cito di Pense wrote:Then there are multiple easy solutions to find, and you can get busy finding the next one.
Users viewing this topic: No registered users and 1 guest