Closed form expression for an infnite product

Question

I am interested in a closed form expression for the limit of the sequence $(a_n)$ where \begin{equation} a_n = \prod_{k=1}^n (1 - \tfrac{c}{k}) \end{equation}

where $c$ is not equal to $1$ and is positive. According to Wolfram Alpha, the sum $\sum_{i=1}^\infty \log(1 - c/n)$ diverges for all $c \neq 0$ which seems to imply that the product does not converge to a non-zero real. But I ran a couple of Matlab tests and the product seems to converge for pretty much all values of $c$ that I tried. Any ideas on how to approach this problem?

Answer 1

One approach that could be made rigorous is the following: Note that $\log(1-x)\simeq -x$ for small positive $x$. You get a harmonic series (from your sum of logarithms) which diverges to $-\infty$, and then $e^{-\infty} = 0$ of course. So your sequence converges to $0$ for every positive $c$.

Answer 2

The equivalent of $a_n$ for $n$ tending to infinity is $$\frac{1}{n^c\Gamma(1-c)}$$