Let $n \ge 2$ be a positive integer and let $x_n$ be a positive real root to the equation $x(x+1)...(x + n) = 1$. Prove that $$x_n <\frac{1}{\sqrt{n! H_n}}$$where $H_n = 1+\frac12+...+\frac{1}{n}$.
Source: 2011 Saudi Arabia Pre-TST November 4.3
Tags: algebra, inequalities
Let $n \ge 2$ be a positive integer and let $x_n$ be a positive real root to the equation $x(x+1)...(x + n) = 1$. Prove that $$x_n <\frac{1}{\sqrt{n! H_n}}$$where $H_n = 1+\frac12+...+\frac{1}{n}$.