Sumproduct For Next Year

Calculus Level 4

\[\large \displaystyle \sum_{i=1}^{\infty} \left ( {\displaystyle \prod_{n=1}^{2017} (i+n)} \right)^{-1} \]

If the value of above expression is in the form \(\dfrac{1}{a \cdot b!}\), where \(a\) and \(b\) are positive integers with \(a<b\), find \(a+b\).

Bonus: Generalise it.

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