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 \(Q\).

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