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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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