Gauss's Hypergeometric Theorem/Examples/Corollary 2 2pi over phi

Example of Use of Gauss's Hypergeometric Theorem

 * $\paren {\dfrac 1 {10^{-1} \times 3 \times 0!} } + \paren {\dfrac 3 {10^0 \times 13 \times 1!} } + \paren {\dfrac {3 \times 13} {10^1 \times 23 \times 2!} } + \paren {\dfrac {3 \times 13 \times 23} {10^2 \times 33 \times 3!} } + \cdots = \dfrac {2 \pi} \phi$

Proof
From Gauss's Hypergeometric Theorem: Corollary 2:


 * $\ds \dfrac 1 a + \dfrac a {\paren {a + 1} 1!} + \dfrac {a \paren {a + 1} } {\paren {a + 2} 2!} + \dfrac {a \paren {a + 1} \paren {a + 2} } {\paren {a + 3} 3!} + \cdots = \dfrac {\pi} {\map \sin {\pi a } }$

Let $a = \dfrac 3 {10}$.

On the :

On the :

Therefore:
 * $\paren {\dfrac 1 {10^{-1} \times 3 \times 0!} } + \paren {\dfrac 3 {10^0 \times 13 \times 1!} } + \paren {\dfrac {3 \times 13} {10^1 \times 23 \times 2!} } + \paren {\dfrac {3 \times 13 \times 23} {10^2 \times 33 \times 3!} } + \cdots = \dfrac {2\pi} \phi$