solving linear equations made difficult

2012-07-17T04:13:26Z

(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).

Answer by Martin Brandenburg for solving linear equations made difficult

2012-07-17T12:24:07Z

Actually this calculation has a formal sense in every ring, by working in the ring of formal power series in $a$ (here $1-a$ is invertible with inverse $\sum_{i \geq 0} a^i$). There are many "pseudo-analytic" proofs in ring theory (one was discussed here). I've made this CW because I cannot answer the question whether this has appeared in the literature, but I am pretty sure that it has.

solving linear equations made difficult - MathOverflow

solving linear equations made difficult

Answer by Martin Brandenburg for solving linear equations made difficult