(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).

  • 11
    $\begingroup$ This seems to be inspired by the standard proof of Picard-Lindelöf (notice the step which can be phrased as an appeal to the Banach fixed point theorem). $\endgroup$ – Qiaochu Yuan Jul 17 '12 at 5:26
  • 11
    $\begingroup$ A silly and obvious observation is that it is of course a cute reversal of the usual hand-waving way to sum the geometric series: if $S=\sum_{k\ge0}ba^k$, we instantly see that $S=b+aS$, so $S=\frac{b}{1-a}$. $\endgroup$ – Vladimir Dotsenko Jul 17 '12 at 8:47
  • 5
    $\begingroup$ Indeed I do not see anything unusual in the computation of the question. For some numerical values, this is also the numeric 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. $\endgroup$ – Pietro Majer Jul 17 '12 at 11:45
  • 8
    $\begingroup$ What Qiaochu says is the key point: the map $f: f(x) = ax +b$ is a contraction mapping on $\mathbb{R}$ as long as $|a| < 1.$ Hence the sequence $(f^{n}(x))$ is Cauchy and converges to the unique fixed point of $f$ for every real number $x.$ The fact that $f$ happens to be semilinear is a secondary issue. $\endgroup$ – Geoff Robinson Jul 17 '12 at 12:05
  • 8
    $\begingroup$ Maybe I'm being dumb, but I can't understand this sudden interest for $2=1+1/2+1/4+1/8+\dots$, at MO! Why not $(a+b)(a-b)=a^2-b^2$ then? $\endgroup$ – Pietro Majer Jul 17 '12 at 21:59

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.

  • 4
    $\begingroup$ Isn't that how one proves that the ring of formal power series is local? $\endgroup$ – Will Sawin Jul 17 '12 at 13:28

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.