Solving for $x$: $1=\frac{1}{x}+\frac{1}{1+\frac{1}{x}}+\frac{1}{1+\frac{1}{1+\frac{1}{x}}}+\cdots$

How can I solve for $x$:

$$1=\cfrac{1}{x}+\cfrac{1}{1+\cfrac{1}{x}}+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{x}}}+\cdots$$

Any clues?

• does the rhs converge for any $x$?
– yoyo
Jun 14 '13 at 18:16
• Some quick calculations lead me to believe that the series does not converge, but I might be wrong. Jun 14 '13 at 18:24
• Each term can be written as $\frac{a+bx}{c+dx}$ where $a,b,c,d$ are Fibonacci numbers. I'll need a little moment to figure out which they are. Jun 14 '13 at 18:26
• @ThomasAndrews $\displaystyle \frac{F_nx+F_{n-1}}{F_{n+1}x+F_n}$ starting at $n=0$ I believe. (typo)
– anon
Jun 14 '13 at 18:27

We deal with a function $x\mapsto \frac{1}{1+\frac{1}{x}}$, which if we take $\frac{1}{x}$ as argument, we should rather write as $x\mapsto \frac{1}{1+x}$.

Iterated, this gives the fixed point iteration for finding a soltion of $\frac{1}{1+x}=x$. And therefore all your terms eventually turn out converge to a solution of $1=(1+x)\ x$, namely $\frac{\sqrt{5}-1}{2}$ (see golden ratio), making the total sum diverge.

Some Mathematica code: • Do the terms ever converge to $-\varphi$ besides $x=-\varphi$?
– anon
Jun 14 '13 at 18:34
• @anon: $|\varphi|\approx 0.618<1$, so the result is positive. There is no square, the $x$ on the right hand side of $\frac{1}{1+x}=x$ is dropped. Jun 14 '13 at 18:42
• @vermiculus: eww, why did you do that? ;P Jun 15 '13 at 0:03
• @NickKidman Hehehehehe Hey man, if people vote it up, it ought to be persistent. It'd be nice to see a formal proof, even though the original equation is a definition for $\phi$, but a table --- while not conclusive --- has immediate and observable value. (What would be better to see is a graph of these data.) Jun 15 '13 at 0:06
• @vermiculus: Okay, I see. Well I've improved on the pic now. On your request, I tried to make a plot, but since starting input like $x=1000$ hops to the tiny $\frac{1}{1001}$ after only one iteration and then stays in a range of $(0,1)$, the plot isn't particularly pretty. I could cancel the first value, but still, not spectacular, if you ask me. Jun 15 '13 at 0:28

$$a_{n+1} = \frac{1}{1+a_n},\ a_0=1/x,\ \sum_{n=0}^{\infty}a_n=1$$

Unfortunately, $a_n$ converges to a non zero constant $\frac{\sqrt{5}-1}{2}$, so that the sum of all $a_n$'s is infinity.