Solving Inequalities with Gamma Function

I am wondering why Mathematica outputs that the following system "cannot be solved with the methods available to Reduce".

$\frac{\Gamma(\frac{1}{2}+n)}{n-1}<\frac{\Gamma(\frac{1}{2}+k)}{k-1}$

with $k, n \in N$. I used the command

Reduce[Gamma[1/2 + n]/(-1 + n) < Gamma[1/2 + k]/(-1 + k) && k > 1 && n > 1, {n}, Integers]

and tried unsuccessfully to solve for $k$ and $n$.

I suppose, I do something improper and Mathematica is capable to solve those kind of inequalities. So, what did I do wrong?

Comments

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