# Reciprocal of One minus x in terms of Gaussian Hypergeometric Function

## Theorem

$\dfrac 1 {1 - x} = {}_2 \map {F_1} {1, p; p; x}$

where:

$x$ and $p$ are real numbers with $\size x < 1$
${}_2 F_1$ denotes the Gaussian hypergeometric function.

## Proof

 $\ds {}_2 \map {F_1} {1, p; p; x}$ $=$ $\ds \sum_{n \mathop = 0}^\infty \frac {1^{\bar n} p^{\bar n} } {p^{\bar n} } \frac {x^n} {n!}$ Definition of Gaussian Hypergeometric Function $\ds$ $=$ $\ds \sum_{n \mathop = 0}^\infty n! \frac {x^n} {n!}$ One to Integer Rising is Integer Factorial $\ds$ $=$ $\ds \sum_{n \mathop = 0}^\infty x^n$ $\ds$ $=$ $\ds \frac 1 {1 - x}$ Sum of Infinite Geometric Sequence, justified as $\size x < 1$

$\blacksquare$