Convergence of series of quotients of gamma functions I am interested in determining if the following series converges:
$$\sum_{k=0}^{\infty} \frac{\Gamma(k+2\delta)}{\Gamma(k+1)} \;,$$
where $\delta > 0$ is a real number and the gamma function is defined by $$\Gamma(\alpha) = \int_0^{\infty} x^{\alpha-1} e^{-x} \, dx \;,$$ for $\alpha > 0$. Any thoughts on this would be much appreciated.

For $\delta>0$ and $k\geq1$ we have $\Gamma(k+2\delta)>\Gamma(k)>0$ and $\Gamma(k+1)>0$, so
$$\frac{\Gamma(k+2\delta)}{\Gamma(k+1)}>\frac{\Gamma(k)}{\Gamma(k+1)}=\frac1k.$$
It follows that the series diverges because the harmonic series diverges.
