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

Solution 1:

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:

Solution 2:

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

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.