Logarithm of One plus x in terms of Gaussian Hypergeometric Function

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Theorem

$\ds \map \ln {1 + x} = x \, {}_2 \map {F_1} {1, 1; 2; -x}$

where:

$x$ is a real number with $\size x < 1$
${}_2 F_1$ denotes the Gaussian hypergeometric function.


Proof

\(\ds x \, {}_2 \map {F_1} {1, 1; 2; -x}\) \(=\) \(\ds x \sum_{n \mathop = 0}^\infty \frac {\paren {1^{\overline n} }^2} {2^{\overline n} } \frac {\paren {-x}^n} {n!}\) Definition of Gaussian Hypergeometric Function
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \paren {n!}^2 \paren {\frac {\paren {2 - 1}!} {\paren {2 + n - 1}!} } \frac {x^{n + 1} } {n!}\) Rising Factorial as Quotient of Factorials, One to Integer Rising is Integer Factorial
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {\paren {n!}^2} {\paren {n + 1}! n!} x^{n + 1}\)
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {\paren {n!}^2} {\paren {n + 1} \paren {n!}^2} x^{n + 1}\) Definition of Factorial
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 1}^\infty \paren {-1}^{n - 1} \frac {x^n} n\) shifting the index
\(\ds \) \(=\) \(\ds \map \ln {1 + x}\) Power Series Expansion for $\map \ln {1 + x}$

$\blacksquare$


Sources