Computing the maximum of a (Classic) function involving factorial I'm trying to compute the global maximum (or an upper bound that does not depends on x) for the expression
$$1/x!\binom{x+n-1}{x},$$
as a function of x. Where $n$ in a positive integer parameter.
An equivalent problem would be finding the maximum of
$$\frac{\Gamma(x+n-1)}{\Gamma(x)^2}$$ 
If it helps, we can assume that x is a natural number and bound the above only for such.
The best case scenario is if someone can point out a known closed form for the maximum.
Other then that, a reasonable tight bound that does not not depends on x would also help.
Thanks.
Looking at f(x+1)/f(x) solves it as one of the comments suggests.
Looking at f(x+1)/f(x) solves it as one of the comments suggests.