Infinite summation

Algebra Level 4

If n=11n(n+1)(n+2)(n+6)=1a \displaystyle \sum_{n=1}^\infty \dfrac1{n(n+1)(n+2)\cdots (n+6)} = \dfrac1a , find a6\dfrac a6.

Hint: Generalize it for n=11n(n+1)(n+2)(n+k)\displaystyle \sum_{n=1}^\infty \dfrac1{n(n+1)(n+2)\cdots(n+k)} .

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