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 left hand side:

\(\ds \)

\(\)

\(\ds \dfrac 1 {\paren {\dfrac 3 {10} } } + \dfrac {\paren {\dfrac 3 {10} } } {\paren {\dfrac 3 {10} + 1} 1!} + \dfrac {\dfrac 3 {10} \paren {\dfrac 3 {10} + 1} } {\paren {\dfrac 3 {10} + 2} 2!} + \dfrac {\dfrac 3 {10} \paren {\dfrac 3 {10} + 1} \paren {\dfrac 3 {10} + 2} } {\paren {\dfrac 3 {10} + 3} 3!} + \cdots\)

\(\ds \)

\(=\)

\(\ds \dfrac 1 {\paren {\dfrac 3 {10} } } + \dfrac {\paren {\dfrac 3 {10} } } {\paren {\dfrac {13} {10} } 1!} + \dfrac {\dfrac 3 {10} \paren {\dfrac {13} {10} } } {\paren {\dfrac {23} {10} } 2!} + \dfrac {\dfrac 3 {10} \paren {\dfrac {13} {10} } \paren {\dfrac {23} {10} } } {\paren {\dfrac {33} {10} } 3!} + \cdots\)

\(\ds \)

\(=\)

\(\ds \paren {\dfrac 1 {10^{-1} \times 3 \times 0!} } + \paren {\dfrac 3 {10^0 \times 13 \times 1!} } + \paren {\dfrac {3 \times 13} {10^1 \times 23 \times 2!} } + \paren {\dfrac {3 \times 13 \times 23} {10^2 \times 33 \times 3!} } + \cdots\)

On the right hand side:

\(\ds \dfrac \pi {\map \sin {\dfrac {3 \pi} {10} } }\)

\(=\)

\(\ds \dfrac \pi {\paren {\dfrac \phi 2} }\)

Sine of Complement: $\map \sin {\dfrac \pi 2 - \dfrac \pi 5} = \cos \dfrac \pi 5$ and Golden Ratio: $\cos \dfrac \pi 5 = \dfrac \phi 2$

\(\ds \)

\(=\)

\(\ds \dfrac {2 \pi} \phi\)

Therefore:

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

$\blacksquare$