Answer

\[ \begin{aligned} R &= \frac{1}{\lim_{m \to \inf} \detm{\frac{1/(2(m+1)+1)!}{1/(2m+1)!}}}\newline &= \frac{1}{\lim_{m \to \inf}\detm{\frac{1}{(2m+3)(2m+2)}}} = \inf \end{aligned} \]

i.e., convergence happens always.