Gauss's Hypergeometric Theorem/Examples/2F1(0.5,0.5;1.5;1)

Example of Gauss's Hypergeometric Theorem

 * $1 + \paren {\dfrac 1 {2^1 \times 3 \times 1!} } + \paren {\dfrac {1 \times 3} {2^2 \times 5 \times 2!} } + \paren {\dfrac {1 \times 3 \times 5} {2^3 \times 7 \times 3!} } + \cdots = \dfrac \pi 2$

Proof
From Gauss's Hypergeometric Theorem:


 * $\map { {}_2F_1} {a, b; c; 1} = \dfrac {\map \Gamma c \map \Gamma {c - a - b} } {\map \Gamma {c - a} \map \Gamma {c - b} }$

where:
 * $\map { {}_2F_1} {a, b; c; 1}$ is the hypergeometric series: $\ds \sum_{k \mathop = 0}^\infty \dfrac { a^{\overline k} b^{\overline k} } { c^{\overline k} } \dfrac {1^k} {k!}$
 * $x^{\overline k}$ denotes the $k$th rising factorial power of $x$
 * $\map \Gamma {n + 1} = n!$ is the Gamma function.

We have:

and:

Therefore:
 * $1 + \paren {\dfrac 1 {2^1 \times 3 \times 1!} } + \paren {\dfrac {1 \times 3} {2^2 \times 5 \times 2!} } + \paren {\dfrac {1 \times 3 \times 5} {2^3 \times 7 \times 3!} } + \cdots = \dfrac \pi 2$