If $$p_n(x)=\prod_{i=1}^{n}(x+i)$$ then what is the order of the power of $x$ (with respect to $n$) with the greatest coefficient for the following polynomial? $$\lim_{n \to \infty} p_n(x)$$
I plugged in some small values of $n$ and saw that the desired power tends to hang within the smaller powers of $x$ (for example, $p_4(x)=x^4 + 10 x^3 + 35 x^2 + 50 x + 24$, so the power of $x$ with the greatest coefficient is $1$), but I don't know enough about infinite polynomials to make any sort of generalization.
Also, I know that the power goes to infinity as $n$ goes to infinity, what I want to know is the order of the power relative to $n$- is it $\sqrt{n}$? $\ln(n)$? $n^{\frac{1}{e}}$? That sort of thing.