="James Propp" <jamespropp@gmail.com> I saw this amusing derivation on the blackboard at MSRI [...]
"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? [...]
Heh, haven't seen it, but that IS amusing. However--just for fun doubtless taking this far more seriously than intended--I really like how it casts the problem as finding a fixpoint rather than just expression shuffling, and how it leverages infinite symbolic objects. I'd even advocate dropping that "(assuming |a| < 1)" and blithely proceed symbolically (trusting in the spirit of Euler) and only lastly go back and ponder what happens when we bind the resulting abstract forms in different concrete domains. For example it shows, as recently discussed here, why binary ...111 = -1 (heh, justifying two's complement arithmetic by analytic continuation?!) The pattern also generalizes to, say, evaluating continued fractions, etc.