harmonic series (idea)

Convergence Of The Rearranged Alternating Harmonic Series

Although the harmonic series doesn't converge, the alternating harmonic series
1 - (1/2) + (1/3) - (1/4) - ...
does, and it converges to the natural logarithm of 2.

This fact can be generalized in an interesting way. If we rearrange the harmonic series so that we take, alternately, r positive terms and s negative ones, the series converges to (1/2) ln (4r/s).

The proof of this, for me, took roughly four pages, but the basic gist of it is this:

• Assign powers of x to each term in the series, so that the exponent of x in each positive term is s times the denominator of that term, and the exponent in each negative term is r times the denominator of that term.
• Rearrange the terms so that all the positive terms and all the negative terms are together, and split the positive terms in half so that all the denominators are even.
• Factor out a (1/2) from the series, and use the Taylor series for ln (1 + x) to simplify what remains.
• Take the limit of the result as x approaches 1 from the left, since it doesn't seem to have a value at 1. Recall that we assigned powers of x to each term, so when x=1, we have our original series.

That part is simple enough. It just takes some slightly clever algebraic manipulation and the knowledge of the Taylor series for ln (1 + x), which is
ln (1 + x) = x - (x²)/2 + (x³)/3 - . . .
The hard part comes with filling in the details. The hardest part is showing that the series is continuous at x=1, so that we can take the limit in the last step and know that it's the same as the actual value of the series at 1. In order to show that a series of functions is continuous, we need to show that it uniformly converges there, so this is not trivial.

This result is slightly less interesting in light of the fact that any conditionally convergent series can be rearranged to converge to any real number, but it is kind of neat nonetheless.

Note: I did a project on this for my Real Analysis class, but this node is not cut and paste. Thanks to David Bressoud of Macalester College for assigning it.