A problem I am faced with is analyzing the boundedness of
$F(k,d,\xi) = \prod_{j=1}^{j=k} {(1+\frac{\xi}{j})^d}$
More specifically, I want to analyze F for large d and k and check when it is bounded.
Progress until now:
(a) For $\xi$ > 0 strictly then as d,k $\rightarrow$ $\infty$ we have F $\rightarrow$ $\infty$. So there is no hope in this regime.
(b) For $\xi$ of $O(1/d)$ and k $\rightarrow$ $\infty$ "slower than"(how do I quantify this?) d $\rightarrow$ $\infty$, we have F $\rightarrow$ $k^{\frac{\xi}{d}}$ and hence F is bounded.
I need to analyze the function for other regimes. My question is how do I go about doing it in a systematic way and what are the results?