Question 81
Let $n \geq 2$ be a positive integer and let $x_{n}$ be a positive real root to the equation $x(x+1) \ldots(x+n)=1$. Prove that
$$
x_{n}\lt \frac{1}{\sqrt{n!H_{n}}}
$$
where $H_{n}=1+\frac{1}{2}+\ldots+\frac{1}{n}$.
$$
x_{n}\lt \frac{1}{\sqrt{n!H_{n}}}
$$
where $H_{n}=1+\frac{1}{2}+\ldots+\frac{1}{n}$.