I'm confused by this problem. Is the answer 1 or 0 OR 1/2? If the

  • #26
D H
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This post contains replies to several recent posts.


Eilenberg-Mazur
Huh? Except for the connection made in very poorly written (and apparently biased) wikipedia article on the Eilenberg-Mazur swindle, there is no connection between the Grandi series and this swindle. The article equates fallacious proofs of 1=0 via the Grandi series to the Eilenberg-Mazur swindle. This is fallacious reasoning in and of itself. For one thing, the Eilenberg-Mazur swindle is valid.

For another, there's nothing special about the Grandi series here. Consider the alternating harmonic series, 1-1/2+1/3-1/4+... This is a convergent series: It converges to ln(2). Now let's rearrange the terms of the series. The algorithm for the rearrangement is a state machine that alternates between two states:
  • In state 1 (the initial state), find the first remaining additive term (e.g., 1, 1/3, 1/5, ...). Pair this with the subtractive term that is half of the first term. Add the sum of the two as a new term in the rearranged series. Eliminate both terms in the original formulation from further consideration.
  • In state 2, find the first remaining subtractive term. Add this term as a new term in the rearranged series and eliminate the found term in the original formation from further consideration.

The rearranged series is (1-1/2)-1/4+(1/3-1/6)-1/8+(1/5-1/10)-1/12+(1/7-1/14)-1/16+... = 1/2-1/4+1/6-1/8+1/10-1/12+1/14-1/16 = 1/2*(1-1/2+1/3-1/4+...). By rearranging the terms in the series we have halved the value of the series! In fact, given any number, it is possible to find a rearrangement of the alternating harmonic series that yields that number as a sum.

The problem here is that alternating harmonic series is a conditionally convergent series; it is not absolutely convergent. This magic of rearranging terms to come up with drastically different sums applies to any conditionally convergent series. What this means is that rearranging terms is invalid for such series. (It is a valid operation for absolutely convergent series.)


This may be elementary but why can't it be solved this way:

S = 1 - 1 + 1 - 1 + 1 - 1 + . . .

S = 0 + 1 - 1 + 1 - 1 + 1 - 1 + . . .

Add
2S = 1

S = 1/2
The problem here is that you are manipulating terms of an alternating series that is not absolutely convergent. It is an illegal operation.


It doesn't work because there is not such S. In other words, it's nonsensical (in the sense of adding an infinite series with another).
There certainly is such an S. You just can't find it using elementary techniques. Cesaro summation, Abel summation, Borel summation, and zeta function regularization all yield the same answer, 1/2, to 1-1+1-1+...


It is perfectly fine if the series converge. It rests on the fact that if a_n --> a, b_n --> b, then a_n+b_n --> a+b.
As noted above, this is not a valid operation on conditionally convergent series.
 
  • #27
disregardthat
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There certainly is such an S. You just can't find it using elementary techniques. Cesaro summation, Abel summation, Borel summation, and zeta function regularization all yield the same answer, 1/2, to 1-1+1-1+....
Of course, I meant that [tex]\lim_{N \to \infty} \sum^N_{n=0} (-1)^n[/tex] does not exist. It should be obvious from the context.


As noted above, this is not a valid operation on conditionally convergent series.
It most certainly is valid. Note that I don't allow a rearrangement of the terms as you propose in your post. It's a matter of adding finite partial sums with well defined terms, something I earlier have stressed. Now I'm talking about converging series, not zeta function regularization or something else.
 
  • #28
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... For one thing, the Eilenberg-Mazur swindle is valid...
that's right, that is what I meant in my post #18:
scientists may debate for centuries ("liar" paradox [true/false]) when they are reluctant to accept the principle that "exception proves the rule"
...For another, there's nothing special about the Grandi series here...
Grandi has some analogy [convergent/divergent] with Russell's http://wikipedia.org./wiki/Present_King_of_France" [Broken] [true/false], if you consider it as the antecedent of Leibniz [a= -2]
[tex]\sum^{\infty}_{n=0} [a=-1]^{n}[/tex]
It ought to be re-written, re-defined.
Is 1,0,0,0,... a series? is it convergent? is it divergent? what is Cesaro-sum?

the sense of post #18 was:
[so expresses logical implication [itex]\Rightarrow[/itex]] : disregardthat has made a valid argument:
... they alternate between 0 and 1, and so will not converge at all...
values alternate [itex]\Rightarrow[/itex] series is not convergent. [[itex]\Rightarrow[/itex]....]
This is a new argument, because swindle [is valid] had already proven that, but had the consequence ("tertium non datur") that series is convergent.

This is a better argument, as it applies to both "possibilities" and we must not wait for another swindle. If it is so, disregardthat has made a swindle-preventing discovery.
 
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  • #29
pwsnafu
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[tex]\sum^{\infty}_{n=0} [a=-1]^{n}[/tex]
You need to define your notation. I've never seen that.

Is 1,0,0,0,... a series? is it convergent? is it divergent? what is Cesaro-sum?
It is convergent sequence, and not a series.
Cesaro is http://en.wikipedia.org/wiki/Ces%C3%A0ro_summation" [Broken].
 
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