Proving It Is The Fun Part!

Algebra Level 5

$\dfrac1{(x+1)(x+2)\cdots(x+999) }$

The partial fraction decomposition of the expression above can be expressed as $\displaystyle \sum_{m=1}^{999} \dfrac {a_m}{x+m}$ for some $a_1, a_2, \ldots , a_{999}$ .

If $\max \{|a_1|, |a_2|, \ldots, |a_{999}|\} = \dfrac{1}{(Q!)^2}$ , find $Q$ .

Notations:

$| \cdot |$ denotes the absolute value function.
$!$ denotes the factorial notation. For example, $8! = 1\times2\times3\times\cdots\times8$ .