For sufficiently large $m$, one can approximate the function
$$f:m\mapsto\frac{\Gamma \left(\frac{m+1}{2}\right)^2}{\Gamma \left(\frac{m}{2}\right) \Gamma \left(\frac{m}{2}+1\right)}$$
using the taylor polynomial with some degree $n$ of $f(z=1/m)$ in the limit $z\to 0^+$, i.e.
$$1-\frac{1}{2 m}+\frac{5}{8 m^2}-\frac{11}{16 m^3}+\frac{83}{128 m^4}-\frac{143}{256 m^5}+\frac{625}{1024 m^6}-\frac{1843}{2048 m^7}+\frac{24323}{32768 m^8}+\ldots$$
However, the radius of convergence at $z\to 0^+$ is $0$, so if $m$ is fixed this approximation doesn't converge (although being very close when using an appropriate degree).
If $m$ is huge and fixed, is there any alternative which will converges to arbitrary precision if sufficiently many iterations/terms/something else are used?