How to find the nature of the following series

Question

I have sum of $$ \frac{n!}{(a+1)(a+2)..(a+n)}$$.I need to check this for convergence I tried writing (a+1)(a+2)..(a+n) as (a+n)!/a! but I don't know what to do next.(where ! is the factorial)

Answer 1

Hint:

First, Look at the cases $a=0$ and $a=1$ what kind of series does the result look like?

Then look at the cases $a\geq 2$ and estimate the upper bound for the expression. Compare the result with the Euler series (aka Basel Problem)

$$\sum_{i=1}^{\infty}\dfrac{1}{n^2}=\dfrac{\pi^2}{6}.$$

What can you conclude from the observation that the sum only contains positive terms and is bounded by a constant number?

Answer 2

$$u_n=\frac{n!}{(a+1)(a+2)\cdots(a+n)}\implies \frac{u_{n+1}}{u_n}=\frac{n+1}{a+n+1}$$

On the other side $$\sum_{n=0}^\infty u_n=\frac{\Gamma (a-1)\, \Gamma (a+1)}{\Gamma (a)^2}$$ which, if $a$ is an integer, reduces to $\frac 1a$.

As zwin commented, "if $a$ is an integer, you can calculate the sum by transforming it into a telescoping series"; this is the real key.