Let $a_n$ be a sequence of real numbers. We can define a formal finite continued fraction as usual
$$[a_0]=a_0,[a_0,a_1]=a_0+\frac{1}{[a_0]},\cdots,[a_0,a_1,\cdots,a_n]=a_0+\frac{1}{[a_1,\cdots,a_n]}.$$
If all $a_n\ge0$, then by Seidel-Stern Theorem and Stern-Stolz Theorem, the formal continued fraction converges if and only if $\sum a_n=\infty$.
What about the sequence with alternative sign: $a_n a_{n+1}<0$ for all $n\ge0$? Is there some conditions that guarantee the well-definiteness and the convergence of $[a_0,a_1,\cdots,a_n]$?
Thanks!
Not a complete answer but Pringsheim’s Theorem says that $|a_i|\ge 2$ is sufficient for convergence. See DLFM 1.12.