I have to study the convergence of the following series:
$\sum_{n\ge1} \frac{ n! } { p (p+1) \cdots (p + n - 1) }\text{ where }p > 0.$
I tried d'Alembert criterion but $\lim_{n\to\infty} \frac {a_{n+1}} {a_n} = 1$ (where $a_n = \frac{ n! } { p (p+1) \cdots (p + n - 1) }$).
Because that limit is $1$ the nature of the series is inconclusive.
Intuitively I can say that the series is convergent because when $p\in\{1,2,n\}$ the sum becomes:
For $p = 1$, $\sum_{n\ge1} \frac{ n! } {1\cdot2\cdots(1+n-1)} = \sum_{n\ge1} \frac{n!}{n!} = n$ is convergent
and
For $p = 2$, $\sum_{n\ge1} \frac{ n! } {2\cdot3\cdots(2+n-1)} = \sum_{n\ge1} \frac{n!}{2\cdot3\cdots(n+1)} = \sum_{n\ge1} \frac{n!}{(n+1)!} =\sum_{n\ge1} \frac{1}{n+1}$ is convergent
and
For $p = n$, $\sum_{n\ge1} \frac{ n! } {n(n+1)\cdots(2n-1)} = \sum_{n\ge1} \frac{(n-2)!}{(n+2)(n+3)\cdots(2n-1)}$ is convergent (d'Alembert)
How do I proof my intuition is a rigorous mathematical way?