This post contains replies to several recent posts.
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.Eilenberg-Mazur
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.)
The problem here is that you are manipulating terms of an alternating series that is not absolutely convergent. It is an illegal operation.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 + . . .
2S = 1
S = 1/2
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 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).
As noted above, this is not a valid operation on conditionally convergent series.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.