Let $a_n>0$ be a decreasing sequence of real numbers and consider the continued fraction $$\cfrac{ 1}{ 1- \cfrac{a_0}{1- \cfrac{a_1}{1- \ddots}}}.$$ Is there any criteria for such continued fractions to converge? The most applicable criterium I have found so far is Worpitzky's theorem which guarantees convergence whenever $\max a_n \leq 1/4.$ However, this result is far more general since it is concerned with complex valued $a_n$.
The specific case we are interested in is finding the domain of $x$ that gives convergence in the above continued fraction with $$a_n = a_n(x) = \frac{2}{(2 + (n+1) x)(2 + (n+2) x)}.$$ So, $a_n =O(n^{-2})$ and $a_{n+1}/a_n \uparrow 1$.