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Infinite series has no meaning using ordinary arithmetic. You need to define it somehow. Usually you define it as the limit of partial sums, but you could use something else, eg Cesaro summation or Abel summation, to allow more series to converge.aguacate wrote:Also, I know that non-absolutely convergent series can be rearranged so as to equal any value you wish. Does this have anything to do with 1 - 2 + 3 - 4 + ...?
aguacate wrote:To show that the sum equals 1/4, they start out assuming that s = 1 - 2 + 3 - 4 + ... exists.
aguacate wrote:Also, I know that non-absolutely convergent series can be rearranged so as to equal any value you wish. Does this have anything to do with 1 - 2 + 3 - 4 + ...?
Hix wrote:aguacate wrote:Also, I know that non-absolutely convergent series can be rearranged so as to equal any value you wish. Does this have anything to do with 1 - 2 + 3 - 4 + ...?
The result you are referring to requires a series which is already known to converge, but not to absolutely converge. So that result cannot be applied to "1-2+3-4+...", which does not converge at all.
aguacate wrote:So the result 1/4 is a sort of average on the partial sums of the series 1 - 2 + 3 - 4 + ... You can see this by looking at the average of say the 51st term (26) and the 50th term (-25) : 1/2. Then look at the average of the 52nd (-26) and 51st term (26): 0. Now we compute the averages of the averages (1/2 and 0): 1/4.
hotaru wrote:aguacate wrote:So the result 1/4 is a sort of average on the partial sums of the series 1 - 2 + 3 - 4 + ... You can see this by looking at the average of say the 51st term (26) and the 50th term (-25) : 1/2. Then look at the average of the 52nd (-26) and 51st term (26): 0. Now we compute the averages of the averages (1/2 and 0): 1/4.
wouldn't you get 0 if you average all of them?
parallax wrote:Does anyone know of a proof that 1 - 2 + 3 - 4 + . . . still converges to 1/4 (in the Cesaro sense) even if you rearrange the terms?
Cesaro sums don't necessarily obey the same rules of addition that we are used to. For example, associativity:
(1 - 2) + (3 - 4) + (5 - 6) + . . . = -1 + -1 + -1 + . . . probably doesn't still sum to 1/4.
These are zeta-regularized sums. I don't know if they are Cesaro-summable.
1 + 1 + 1 + 1 + . . . = -1/2
1 + (1+1) + (1+1+1) + . . . = 1 + 2 + 3 + . . . = -1/12
aguacate wrote:parallax wrote:Does anyone know of a proof that 1 - 2 + 3 - 4 + . . . still converges to 1/4 (in the Cesaro sense) even if you rearrange the terms?
Cesaro sums don't necessarily obey the same rules of addition that we are used to. For example, associativity:
(1 - 2) + (3 - 4) + (5 - 6) + . . . = -1 + -1 + -1 + . . . probably doesn't still sum to 1/4.
These are zeta-regularized sums. I don't know if they are Cesaro-summable.
1 + 1 + 1 + 1 + . . . = -1/2
1 + (1+1) + (1+1+1) + . . . = 1 + 2 + 3 + . . . = -1/12
Doesn't 1 - 2 + 3 - 4 + . . . diverge in the cesaro sense? I thought it was the Abel sum that gives it 1/4.
aguacate wrote:To show that the sum equals 1/4, they start out assuming that s = 1 - 2 + 3 - 4 + ... exists. Since it doesn't, their manipulations don't seem to be valid. I mean you could say:
Let x be infinity
infinity plus anything is just infinity
x + 3 is infinity
x + 9 is infinity
x + 3 = x + 9
3 = 9
or something along those lines.
Also, I know that non-absolutely convergent series can be rearranged so as to equal any value you wish. Does this have anything to do with 1 - 2 + 3 - 4 + ...?
infinigesimal wrote:What I'm very uncomfortable with the treatment of the zeta-regularaized sums. The fact that
1 + 1 + 1 + ... = -1/2
doesn't agree with
1 + 2 + 3 + ... = -1/12
is perfectly fine, from the POV that it equality would require the "infinite associativity" which we already know is not guaranteed, But it seems to me that the latter should at least be the square of the former. And the pseudo-equation
11/12 = 1 + -1/12 = 1 + (1 + 1 + ...) = 1 + 1 + 1 + ... = -1/12
shows that you can't even split off a single term if you want well-defined results: evaluating expressions in terms of partial sums breaks things. Then, the meaning of any single addition is dependent on an infinite amount of context to its' left or right. That just ain't right.
Call these zeta continuations what you will, but calling it an infinite summation strikes me as pointless.
archgoon wrote:Can anyone give me a somewhat natural example of how a cesaro sum might occur in solving a problem? Preferably the example would involve "wanting to sum a series" but not being able to unless you cesaro sum.
archgoon wrote:Can anyone give me a somewhat natural example of how a cesaro sum might occur in solving a problem? Preferably the example would involve "wanting to sum a series" but not being able to unless you cesaro sum.
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