(Note: This is a what's-in-the-literature question, not a what's-mathematically-true question, but I believe both are considered valid kinds of MathOverflow question.)

I saw this amusing derivation on the blackboard at MSRI a few months ago (I'm paraphrasing and reformatting slightly, though my attempts at formatting may not work as intended):

"*Problem*: Solve $x = ax + b$ for $x$.

*Solution*:
$$x = a(ax+b) + b = a^2 x + ab + b = a(a(ax+b)+b)+b= a^3 x + a^2 b + ab + b = \cdots$$
(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.

It would be embarrassing to publish this and then receive letters saying "This argument appears almost word-for-word in Littlewood's Miscellany" (or something like that).

thenumeric solution (e.g. having to evaluate $x=8/93$, in elementary school they learn, or used to learn, to write $(100-7)x=8$ or $x=0.08+0.07x=0.08+ 0.07(0.08+0.07x) $ and get $x\sim $0,086. Also, it is also how they learn to switch from fractions to decimal expansions, and conversely, in the representations of rational numbers. – Pietro Majer Jul 17 '12 at 11:45