Power Series Expansion for Logarithm of x/Formulation 2

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

Proof
From the corollary to Power Series Expansion for $\map \ln {1 + x}$:

Let $z = \dfrac 1 {1 - x}$.

Then:

Then we have:
 * $\displaystyle \lim_{x \mathop \to 1^-} \dfrac 1 {1 - x} \to +\infty$

and:
 * $\displaystyle \lim_{x \mathop \to -1^+} \dfrac 1 {1 - x} \to \frac 1 2$

Thus when $x \in \openint {-1} 1$ we have that $z \in \hointr {\dfrac 1 2} \to$.

Thus, substituting $z$ for $\dfrac 1 {1 - x}$ in $(1)$ gives the result.