# Series of ln(x) for x>=1

• MHB
• fluid_mechanics

#### fluid_mechanics

I need to develop $\mathrm{ln}(x)$ into series, where $x \geq 1$, and I don`t know how? In literature I only found series of $\mathrm{ln}(x)$, where:

1. $|x-1| \leq 1 \land x \neq 0$, $\,\,\,\,\, \mathrm{ln}(x) = x - 1 - \dfrac{(x-1)^2}{2} + ...$

2. $|x| \leq 1 \land x \neq -1$, $\,\,\,\,\, \mathrm{ln}(x+1) = x - \dfrac{x^2}{2}+ ...$

My problem is problem in area of fluid dynamics, and $x$ is non-dimensional coordinate and it signifies radial coordinate of annular tube (it starts in the center of the tube). At the wall of inner tube $x=1$, and at the wall of outer tube it only can be larger (and values are not limited), because of that I need to fulfill a condition $x \geq 1$, for developing $\mathrm{ln}(x)$ into series.

Write series for $$\displaystyle \ln (1+x)$$ and $$\displaystyle \ln (1-x)$$. Then subtract second from the first one and you'll obtain the series for $$\displaystyle \ln \left( \frac{1+x}{1-x} \right)$$, which gives you one possible series.