It was shown here Moments of Geometric Random Variable that if $X$ is a geometric random variable, i.e. it represents the number of consecutive failures before you get the first success, where the success probability is $\rho$, there exists absolute constants $C_k$ such that $$ E(X^k)\leqslant\frac{C_k}{\varrho^k}, $$
Is it possible to find this constant, preferably the smallest possible?