Given $a,b\in\mathbb C$, let us construct the following sequence:
$\begin{align} a+b&=a+b\\ \cfrac a{a+b}+\cfrac b{a+b}&=1\\ \cfrac a{\cfrac a{a+b}+\cfrac b{a+b}}+\cfrac b{\cfrac a{a+b}+\cfrac b{a+b}}&=a+b\\ \cfrac a{\cfrac a{\cfrac a{a+b}+\cfrac b{a+b}}+\cfrac b{{\cfrac a{a+b}+\cfrac b{a+b}}}}+\cfrac b{\cfrac a{{\cfrac a{a+b}+\cfrac b{a+b}}}+\cfrac b{{\cfrac a{a+b}+\cfrac b{a+b}}}}&=1\\ &\vdots\\ L(a,b)&=\;? \end{align}$
Since the sequence keeps oscillating between only two numbers, the limit $L(\cdot\,,\cdot)$ doesn't seem to exist, but I saw a claim that $L(1,3)=2$. This could be a hint that it converges to $(a+b)/2$ or $\sqrt{a+b}$ …
So, what is going on here?