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:
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.