Dougall's Hypergeometric Theorem/Examples/4F3(1.25,0.5,0.5,0.5;0.25,1,1;-1)

Example of Use of Dougall's Hypergeometric Theorem

 * $1 - 5 \paren {\dfrac 1 2}^3 + 9 \paren {\dfrac {1 \times 3} {2 \times 4} }^3 - 13 \paren {\dfrac {1 \times 3 \times 5} {2 \times 4 \times 6} }^3 + \cdots = \dfrac 2 \pi$

Proof
From Dougall's Hypergeometric Theorem: Corollary 6:


 * $\ds \map { {}_4 \operatorname F_3} { { {\dfrac n 2 + 1, n, n, n} \atop {\dfrac n 2, 1, 1} } \, \middle \vert \, -1} = \dfrac {\map \sin {\pi n} } {\pi n } $

where:
 * $\ds \map { {}_4 \operatorname F_3} { { {\dfrac n 2 + 1, n, n, n} \atop {\dfrac n 2, 1, 1} } \, \middle \vert \, -1}$ is the generalized hypergeometric function of $1$: $\ds \sum_{k \mathop = 0}^\infty \dfrac { \paren {\dfrac n 2 + 1}^{\overline k} \paren {n^{\overline k} }^3 } { \paren {\dfrac n 2}^{\overline k} \paren {1^{\overline k} }^2 } \dfrac {\paren {-1}^k} {k!}$
 * $x^{\overline k}$ denotes the $k$th rising factorial power of $x$
 * $\map \Gamma {n + 1} = n!$ is the Gamma function.

On the we have:

On the we have:
 * $\ds \dfrac {\map \sin {\dfrac \pi 2} } {\dfrac \pi 2 } = \dfrac 2 \pi$

Therefore:
 * $1 - 5 \paren {\dfrac 1 2}^3 + 9 \paren {\dfrac {1 \times 3} {2 \times 4} }^3 - 13 \paren {\dfrac {1 \times 3 \times 5} {2 \times 4 \times 6} }^3 + \cdots = \dfrac 2 \pi$