Fix $a,d\in \mathbb{R}$ an consider the arithmetic sequence $x_n = a,a+d,a+2d,a+3d,...$ ($x_n = a + (n-1)d $ for each $n$). Now consider $$ A_n = \frac{x_1 + x_2 + \cdots + x_n}{n} \quad \text{ and }\quad G_n = \sqrt[n]{x_1x_2...x_n}$$ I want to compute the limit of $X_n = G_n / A_n$. Note that for the case $a=d=1$, $G_n = \sqrt[n]{n!}$ and $A_n = (n+1)/2$. Using Stirling formula, it yields $\lim X_n = 2e^{-1}$. My work for the general case:
(1) Clearly, by geometric-arithmetic mean inequality, $X_n \leq 1$.
(2) Using the formula for the sum of the terms of an arithmetic sequence, $A_n = \frac{2a+(n-1)d}{2}$
(3) Using a convenient property of Gamma function, $x_1\cdot x_2 \cdots x_n = d^n \frac{\Gamma (a/d + n)}{\Gamma (a/d)} \Rightarrow G_n = d \left(\frac{\Gamma (a/d + n)}{\Gamma (a/d)} \right)^{1/n}$
(4) Lastly, I compute $G_n / A_n$ and I applied $\Gamma (n + a/d) \sim \Gamma (n) n^{a/d}$ as $n\to \infty$. But I got stucked. After some computations, it seems that the limit is infinity... but we know this is impossible since we found a case with finite limit $(a=d=1)$ or, even worse, we know $X_n$ is bounded!. Can anyone help me solving this general case? Thanks in advance.