**Moderator's edit**:

If the value of \( \displaystyle \sum_{m=1}^{26} \left [\sum_{x=0}^\infty \prod_{n=1}^{m+1} \dfrac1{x+n} \right ]^{-1} \) can be expressed as \( \omega! - 1\), find \( \sqrt[3]{\omega} \).

\(If\\ \\ \sum _{ x=0 }^{ \infty }{ } \prod _{ n=1 }^{ 2 }{ \frac { 1 }{ (x+n) } } =\quad A\\ \sum _{ x=0 }^{ \infty }{ } \prod _{ n=1 }^{ 3 }{ \frac { 1 }{ (x+n) } } =\quad B\\ ..............\\ \sum _{ x=0 }^{ \infty }{ } \prod _{ n=1 }^{ 27 }{ \frac { 1 }{ (x+n) } } =\quad Z\\ \quad \\ then\quad \frac { 1 }{ A } \quad +\quad \frac { 1 }{ B } \quad ..........\quad +\quad \frac { 1 }{ Z\quad } \quad can\quad be\quad expressed\quad as\quad \varpi !-1\\ \\ find\quad \sqrt [ 3 ]{ \varpi } \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \)

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