(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).



solving linear equations made difficult
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](https://mathoverflow.net/questions/31595/how-would-you-solve-this-tantalizing-halmos-problem)). 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.


