"Find the sum of series:∑n=0∞(4n)!(4n+4)!" was answered by our community

Taylor Haines

Answered question

2022-02-23

Find the sum of series:

\sum_{n = 0}^{\infty} \frac{(4 n)!}{(4 n + 4)!}

chezpepina87j

Beginner2022-02-24Added 6 answers

\sum_{n \geq 0} \frac{(4 n)!}{(4 n + 4)!} = \sum_{n \geq 0} \frac{Γ (4 n + 1)}{Γ (4 n + 5)}

= \sum_{n \geq 0} \frac{1}{6} B (4 n + 1, 4)

= \frac{1}{6} \sum_{n \geq 0} \int_{0}^{1} x^{4 n} {(1 - x)}^{3} d x

= \frac{1}{6} \int_{0}^{1} \frac{{(1 - x)}^{2}}{(1 + x^{2}) (1 + x)} d x

= \frac{\ln 2}{4} - \frac{π}{24}

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