Prove that the positive root of
x(x + 1)(x + 2) · · · (x + 1981) = 1
is less than 1/1981!
Let $\displaystyle f(x)=x(x+1)(x+2)\ldots (x+1981)-1$
$\displaystyle f(0)=-1$
$\displaystyle f\left(\frac{1}{1981}\right)=\frac{1}{1981}\left(1 +\frac{1}{1981}\right)\left(2+\frac{1}{1981}\right )\ldots\left(1981+\frac{1}{1981}\right)-1=$
$\displaystyle =\left(1+\frac{1}{1981}\right)\ldots\left(1+\frac{ 1}{1981^2}\right)-1>0$
(the factors of the product are greater than 1, so the product is greater than 1)
f is continuos, so exists $\displaystyle x_0\in\left(0,\frac{1}{1981}\right)$ such that $\displaystyle f(x_0)=0$