Finding k for Gamma Function Convergence

[bomba923]

For what values of k would
\mathop {\lim }\limits_{x \to \infty } \frac{{\Gamma \left( {kx + 1} \right)}}<br /> {{x^{kx} }}
converge?

 

[CRGreathouse]

Certainly for k < 0 it diverges, since gamma is ill-behaved there. For k > e it also diverges by Stirling's approximation. There's the easy part! I'll have to think about the remaining (0, e]. (It obviously converges for k = 0.)