Rearrangement of conditionally convergent series?

Question

Please show me the most simple and instructive example of the rearrangement of a conditionally convergent series so that it converges to an arbitrary real number A as well as to plus and minus infinity.

James: thanks, although the link you gave me is just one example and it does not show how to rearrange the series into an arbitrary real number A. I'm still waiting for a really good answer to my question.

Answer 1

Since the series is conditionally convergent, the terms go to zero, but the series of positive terms diverges to infinity as does the series of negative terms. Now, given finite A, add up the first few positive terms until the sum is more than A and is the first such that is more than A. Now subtract off enough negative terms so that the sum is less than A, now add on enough positive terms so that the sum is more than A, etc, etc. At each stage, make sure you only go as far as needed to get the sum just the other side of A. This is always possible because the series of positive and negative terms both diverge. But now, the partial sums converge to A because the terms of the original series go to zero.

If A is infinite, take positive terms to add to more than 2, then negative terms to go less than 1, then positive terms to go more than 4, then negative terms to go less than 3, then 5, then 4, etc...

Answer 2

An occasion of a conditionally convergent sequence that has rearrangements on a line a+bt, with t variable, take a(2n-a million)=-(-a million)^n(a million/n) a(2n)=i/2^n Then those are in 2 perpendicular guidelines and sum to ln(2) + i The alternating harmonic would be rearranged to offer arbitrary values and for that reason the result. the different form, which could have words with out any in parallel guidelines, is extra diffused. i'm useful you could chop up the plane into n sectors from 0, declare there are an countless style of words in one in all them that sum to infinity, with others someplace else canceling. chop up the sectors in a million/2 and proceed. the result would be all in parallel direction or a minimum of two non-parallel. With 2, the entire plane outcomes. i think of the partial sums can save on with an arbitrary direction with a width this is growing to be arbitrarily narrow, yet useful, with in basic terms a finite style of sums excluded.

Answer 3

I wasn't even aware of rearrangement resulting in a change in the value that a series converges to. Sort of makes you wonder about the whole idea! :)

Answer 4

I don't have an example concerning divergence, but here is an example of converging to a different value: