Integration Problem

Calculus Level 3

\[\large 100 \times \dfrac{1-\frac{9}{11}+\frac{9\cdot8}{2!}\frac{1}{21}-\frac{9\cdot8\cdot7}{3!}\frac{1}{31}+\sum_{m=4}^9(-1)^m\frac{\binom{9}{m}}{10m+1}}{1-\frac{10}{11}+\frac{10\cdot9}{2!}\frac{1}{21}-\frac{10\cdot9\cdot8}{3!}\frac{1}{31}+\sum_{n=4}^{10}(-1)^n\frac{\binom{10}{n}}{10n+1}} = \, ? \] \[\] Clarification: The numerator and the denominator contain the general term of summation after explicit 4 terms.

Notation: \(!\) denotes the factorial notation. For example, \(8! = 1\times2\times3\times\cdots\times8 \).

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