Infinite summation

Algebra Level 4

If \( \displaystyle \sum_{n=1}^\infty \dfrac1{n(n+1)(n+2)\cdots (n+6)} = \dfrac1a \), find \(\dfrac a6\).

Hint: Generalize it for \(\displaystyle \sum_{n=1}^\infty \dfrac1{n(n+1)(n+2)\cdots(n+k)} \).

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