Proving It Is The Fun Part!

Algebra Level 5

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

The partial fraction decomposition of the expression above can be expressed as m=1999amx+m\displaystyle \sum_{m=1}^{999} \dfrac {a_m}{x+m} for some a1,a2,,a999a_1, a_2, \ldots , a_{999}.

If max{a1,a2,,a999}=1(Q!)2\max \{|a_1|, |a_2|, \ldots, |a_{999}|\} = \dfrac{1}{(Q!)^2} , find QQ.

Notations:

  • | \cdot | denotes the absolute value function.
  • !! denotes the factorial notation. For example, 8!=1×2×3××88! = 1\times2\times3\times\cdots\times8 .
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