Power Series Expansion for Logarithm of x

Theorem

$\ds \ln x$

$=$

$\ds 2 \paren {\sum_{n \mathop = 0}^\infty \frac 1 {2 n + 1} \paren {\frac {x - 1} {x + 1} }^{2 n + 1} }$

$\ds $

$=$

$\ds 2 \paren {\frac {x - 1} {x + 1} + \frac 1 3 \paren {\frac {x - 1} {x + 1} }^3 + \frac 1 5 \paren {\frac {x - 1} {x + 1} }^5 + \cdots}$

valid for all $x \in \R$ such that $-1 < x < 1$.

$\ds \ln x$

$=$

$\ds \sum_{n \mathop = 1}^\infty \dfrac 1 n \paren {\frac {x - 1} x}^n$

$\ds $

$=$

$\ds \frac {x - 1} x + \frac 1 2 \paren {\frac {x - 1} x}^2 + \frac 1 3 \paren {\frac {x - 1} x}^3 + \cdots$

valid for all $x \in \R$ such that $x \ge \dfrac 1 2$.