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.
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