Rearranging the Alternating Harmonic Series
Terms in the alternating harmonic series (or any other conditionally convergent series) can be rearranged to achieve any desired real number as the sum. This Demonstration allows you to control the target sum and various options to see (part of) the rearranged series, the partial sums, and the pattern of sign changes. By default, the target is set to .
Snapshot 1: making the alternating harmonic series disappear: rearranging terms to achieve a sum of zero leads to a simple and not very well-known identity for 0
Snapshot 2: convergence to 0.12345
Snapshot 3: going negative: the rearranged series will start with negative terms
It turns out that a limiting ratio of positive to negative terms in the partial sums will result in a sum .
Conversely, to achieve a sum , the desired ratio should approach .
For details, see these references:
E. Packel and S. Wagon, "Rearrangement Patterns for the Alternating Harmonic Series," Mathematica in Education, 3(2), 1994 pp. 5–10.
C. C. Cowen, K. R. Davidson, and R. P. Kaufman, "Rearranging the Alternating Harmonic Series," American Mathematical Monthly, 87, 1980 pp. 817–819.