While this method might look strange for regular numbers, I believe that there are some algorithms for _matrices_ that work pretty much like this. At 06:09 PM 6/20/2012, James Propp wrote:
I saw this amusing derivation on the blackboard at MSRI a couple of months ago (I'm paraphrasing and reformatting slightly):
"Problem: Solve x = ax + b for x. Solution: x = a(ax+b) + b = a^2 x + ab + b = a(a(ax+b)+b)+b= a^3 x + a^2 b + ab + b ... = (assuming |a| < 1) lim_{n \rightarrow \infty} a^n x + b sum_{i=0}^{\infty} a^i = 0 + b/(1-a). This also holds by analytic continuation for all a neq 1."
Has anyone seen this before? I took a photograph of the blackboard, and I am inclined to submit it to Mathematics Magazine, but first I want to know the provenance.
Curt McMullen was in residence at MSRI at the time, and he seemed a likely culprit, but when I pointed it out to him he seemed amused, and he denied authorship, so I don't have any suspects at present.
Jim Propp