Convergence of $\sum_{n=1}^{\infty}\frac{\frac{2}{3}(1+\frac{2}{3})\ldots(\frac{2}{3}+n)}{(1+\frac{3}{2})\ldots(\frac{3}{2}+n)}.$ 
Examine the convergence of the series:
$$\sum_{n=1}^{\infty}\frac{\frac{2}{3}(1+\frac{2}{3})\ldots(n+\frac{2}{3})}{(1+\frac{3}{2})\ldots(n+\frac{3}{2})}.$$

Attempt. Using ratio test for $a_n:=\frac{\frac{2}{3}(1+\frac{2}{3})\ldots(n+\frac{2}{3})}{(1+\frac{3}{2})\ldots(n+\frac{3}{2})}>0$, we have:
$$\frac{a_{n+1}}{a_n}=\frac{\frac{2}{3}+n+1}{\frac{3}{2}+n+1}\to 1,$$
so we do not conclude. Also $\sqrt[n]{(1+a)\ldots(n+a)}\to +\infty$ for $a>0$, so the root test also is inconclusive.
Thanks for the help.

In terms of the Gamma and Beta functions we are discussing
$$ \sum_{n\geq 1}\frac{\Gamma\left(n+\frac{5}{3}\right)/\Gamma\left(\frac{2}{3}\right)}{\Gamma\left(n+\frac{5}{2}\right)/\Gamma\left(\frac{5}{2}\right)}=\underbrace{\frac{\Gamma(5/2)}{\Gamma(2/3)\Gamma(5/6)}}_{K}\sum_{n\geq 1}B(n+5/3,5/6) $$
or
$$ K \sum_{n\geq 1}\int_{0}^{1} x^{n+2/3}(1-x)^{-1/6}\,dx=K\int_{0}^{1}x^{5/3}(1-x)^{-7/6}\,dx $$
which is clearly divergent.
