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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.