Suppose we have the following series:
$$ \sum_{n=1}^{\infty}\dfrac{p(p+1)\cdots(p+(n-1))}{n!}z^n $$ where $p\in \mathbb{N}$ and $z\in \mathbb{C}$. I've applied the ratio test finding an easy limit, but, how can I apply the root test for this series? Any help would be appreciated.
Let $$a_n=\prod_{k=0}^{n-1}\frac {p+k}{n!}=p\frac{ (p+1)_{n-1}}{n!}$$ where appears Pochhammer symbol. Now, considering the asymptotics of the logarithm $$\log(a_n)=(p-1)\log(n)+\log \left(\frac{p}{\Gamma (p+1)}\right)+O\left(\frac{1}{n}\right)$$ Neglecting the constant term $$\frac 1 n \log(a_n)=(p-1)\frac{\log(n)} n+O\left(\frac{1}{n^2}\right)$$ $$\sqrt[n]{a_n}=\exp\left(\frac 1 n \log(a_n) \right)\sim n^{(p-1)/n}$$