Radius of convergence of continued fraction with decreasing coefficients

Question

It is well known that the Catalan numbers have generating function $$\frac{2}{1+ \sqrt{ 1- 4z} } = \cfrac{1}{1-\cfrac{z}{1- \cfrac{z}{1-\ddots} } }.$$ This has radius of convergence $1/4$. Is it possible to compute the radius of convergence of a continued fraction in which the $z$ coefficients decrease to $0$? For example, $$\cfrac{1}{1-\cfrac{a_1z}{1- \cfrac{a_2z}{1- \cfrac{a_3z}{1- \ddots} }} }$$ with $a_i \downarrow 0$. We would be interested in any concrete example in which the radius is computable. For instance, a simple case is $a_i = 1/i$.

Answer 1

For your specific case (with a minor change for a simpler formula), we have $$f(z):=\cfrac{1}{1-\cfrac{z/2}{1-\cfrac{z/3}{1-\cfrac{z/4}{1-\ddots}}}}=\int_0^1(1-x)^{-z}e^{-zx}\,dx$$ with the radius of convergence $1$ (for your original CF, the RoC $r$ is the solution of $rf(r)=1$).