09.21.05
Rearranging the Alternating Harmonic Series
One of my favorite theorems in calculus is Riemann’s theorem on conditionally convergent series. To remind you, the theorem says that a conditionally convergent series (i.e. a series that converges only because of the presence of signs, and the cancellation that follows) can be rearranged so that it converges to our favorite real number! What is so wonderful about it is that the statement is unbelievably surprising: The same set of numbers can be made to add up to any number we want, simply by placing them in a different order! On the other hand the proof, which you can find on almost any decent Calculus book, is really not that hard. The brilliance is in figuring out that such a statement could possibly be true.
So when we teach this we always stop by mentioning the theorem, and as a special case mention (perhaps) what happens to the alternating harmonic series, and how it can be rearranged to add up to 
instead of 
, and we always stop there.
I recently found out though, that we can do a lot more with it, without much more effort. One can easily obtain a large class of numbers from the alternating harmonic series, as the following theorem from “Rearranging the Alternating Harmonic Series” by C. C. Cowen, K. R. Davidson and R. P. Kaufman from the American Mathematical Monthly, Vol 87, No 10, Dec 1980 demonstrates:
Suppose we have a rearrangement of the alternating harmonic series, and let
denote the number of positive terms in the first n terms of the rearrangement, and suppose thatexists (possibly infinity). Then, and only then, the rearranged series converges, to the number.
This is really remarkable and very explicit, and can be used in lots of different ways, for instance by having students perform numerical experiments to verify it, (for instance if you place the terms as four positives followed by a negative and so on, the you get 0), or having them read the paper, which is pretty accessible. The proof itself is only one third of a page, and not too scary. Next time I’m teaching series, I am definitely mentioning this!
Later
