A limit with $((n-1)!)^{1/(n-1)}$ and other roots of factorials

Question

How to prove that the following limit is positive? $$ \lim_{n \to \infty}\left(((n-1)!)^{1/(n-1)}-2\left(\frac{((n-1)!)^3}{(2n-2)!}\right)^{1/(n-1)}\right) >0,$$ where $ n\in \mathbb Z, n>1 $

Answer 1

Actually, if you prove that it converges, the condition : $L > 0$ is equivalent to :

$U_n >0$, starting a certain value of n, and $U_n$ being the term you're studying.

Now $U_n >0 \Longleftrightarrow (n-1)! > 2^{n-1}\cdot\dfrac{{(n-1)!}^3}{(2n-2)!}$ since you have positive term

$$U_n >0 \Longleftrightarrow 1 > 2^{n-1}\cdot\frac{{(n-1)!}^2}{(2n-2)!}$$

$$(2n-2)! = (2(n-1))! = (n-1)!n(n+1)\cdots(2n-2) $$

$$ \begin{align} V_n & =\ln\left( \frac{(2n-2)!}{2^{n-1}\cdot(n-1)!^2}\right) = \ln\left( \frac{n (n+1)\cdots(2n-2)}{2^{n-1}\cdot(n-1)!}\right) \\[8pt] & = \sum_{k=1}^{n-1}{(\ln(k+n-1) - \ln(k))} -(n-1)\ln(2) \\[8pt] & = \sum_{k=1}^{n-1}{\left(\ln\left(1+\frac{n-1}{k}\right)\right)} -(n-1)\ln(2) \end{align} $$

$$0 < k < n \Longrightarrow 1 + \frac{n-1}{k} > 2 \Longrightarrow \ln\left(1 + \frac{n-1}{k}\right) > \ln(2)$$ with equality when $k=n-1$.

Sum this from $k=1$ to $(n-2)$ and we get: $$V_n >0 \Longrightarrow (2n-2)! > 2^{n-1}\cdot(n-1)! \Longrightarrow U_n >0$$

This is true for $n>2$, so your limit $L$ is $>0$.