Hilariously Hideous Complex Fractions

Our problem sets so far have included two hilariously hideous complex fractions (both problems I stole from The Art of Problem Solving):

\[\left(i-i^{-1}\right)^{-1}\quad\text{aka}\quad \frac{1}{\,\, i-\frac1i \,\,}\]

and

\[\frac{1}{\,\,1+\frac{1}{\,\,1-\frac{1}{1+i}\,\,}\,\,}\]

These, at least to me, have begotten the question: what if we keep going? What if we nest these sorts of fractions inside fractions inside fractions inside fractions, on and on, ad infinitum? I’m curious because sometimes when we do things an infinite number of times, we get just a finite number. Maybe you’ve seen the cool thing that happens when we add together \(\frac12\), \(\frac14\), \(\frac18\), and all the successive powers of \(2\): we get just \(2\)!

\[1 + \frac12 + \frac14 + \frac18 + \frac{1}{16} + \quad\cdots \quad = 2\]

Sometimes when we do things an infinite number of times, we get cool patterns, like if we raise \(-1\) or \(i\) to powers:

\[(-1)^n = +1, -1, +1, -1, +1, -1, \cdots\]

\[i^n = +1, +i, -1, -i, +1, +i, \cdots\]

So cool stuff can happen when we do things an infinite number of times. What about here? Like, for example, let’s take this second complex fraction. There’s an innermost “\(1+i\)”, and outside of that, we’re just alternatingly adding and subtracting it from \(1\) and reciprocating. So what if we do that yet another time?

\[\frac{1} { 1 - {\color{blue} \frac{1}{\,\,1+\frac{1}{\,\,1-\frac{1}{1+i}\,\,}\,\,}}\,\,}\]

(I put the previous fraction in blue there.) What if we do that again?

\[\frac{1}{1 + {\color{red}\frac{1} { 1 - {\color{blue} \frac{1}{\,\,1+\frac{1}{\,\,1-\frac{1}{1+i}\,\,}\,\,}}\,\,}}\,\,}\]

Et cetera! What happens? What do we get? If we keep fractioning forever and ever, is there a pattern? Does this converge on a single number? (Converge is the fancy math word!) Is it just chaos?

I genuinely don’t know! I’m going to try to figure it out this weekend, too. I hope it’s something cool (and that it’s something that I can figure out!) But if it’s not, that’s OK, too…

(If you want to try something else cool that’s similar—like, e.g., see what happens with the first fraction problem if we keep nesting it in similar fractions—please feel free to! We’re just asking questions and playing with the math…)