What are some references and material that are devoted to studying series of the form
$x_1+\frac1{x_1}+\frac{1}{x_1+\frac1{x_1}}+\dotsc$ where $x_1>0$ is given?
For instance, my (non math) friend just wrote down (somewhat as a "joke") the series
$(1+1/2)+\frac1{1+1/2}+\frac{1}{1+1/2+\frac1{1+1/2}}+\dotsc$
and I am honestly baffled by it. (They actually wrote down $$\frac{1}{(1+1/2)+\frac1{1+1/2}+\frac{1}{1+1/2+\frac1{1+1/2}}+\dotsc}$$ but I figured I would just study the series itself. But would this be related to continued fractions? It doesn't look quite the same in form.)
How would one study the convergence or divergence of the given series (or series of the same form)? I've only encountered sequences defined recursively, say the one to approximate $\sqrt{x}$, or $\sqrt{p+\sqrt{p+\dotsc}}$, or $\sqrt{2}^{{\sqrt{2}}^{\dotsc}}$ and these are relatively easy to handle.
I've tried to study the $n$-th term $x_n=\frac{1}{x_1+x_2+\dotsc+x_{n-1}}$ where $x_1=(1+1/2)$ and the partial sums $s_n=x_1+\dotsc+x_n$ to see if I could say something about the convergence of the sequence $(x_n)$ or $(s_n)$ but the recurrence relation makes it too messy for me to see anything clearly about their behaviors. Thanks in advance for looking at this (and I really hope I'm not missing something obvious here...)
(While writing this question this popped up as related and it has the (edit!) same recurrence .)
Note that the sequence satisfies the recurrence relation $x_{n+1} = x_n + \frac1{x_n} = \frac{x_n^2 + 1}{x_n}$ or $x_{n+1}x_n = x_n^2 + 1$. So if it converges, the limit $x$ must satisfy $$x^2 = x^2 + 1$$ which cannot be. Therefore the sequence cannot converge.
Note that the sequence is always positive, so $1/x_n > 0$ and $x_{n+1} > x_n$, so the sequence is increasing. Increasing + doesn't converge $\implies$ diverges to $\infty$.