Natural logarithm/Taylor series via derivative in 1/Example

We would like to determine the Taylor series of the natural logarithm in the point ${}1$. The derivative of the natural logarithm equals ${}1/x$, due to fact. This function has the power series expansion

${}{\frac {1}{x}}=\sum _{k=0}^{\infty }(-1)^{k}(x-1)^{k}\,,$

due to fact (which converges for ${}\vert {x-1}\vert <1$). Therefore, because of fact, the power series expansion of the natural logarithm is

$\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}}{k}}(x-1)^{k}.$

Setting ${}z=x-1$, we may write this series as

$z-{\frac {z^{2}}{2}}+{\frac {z^{3}}{3}}-{\frac {z^{4}}{4}}+{\frac {z^{5}}{5}}-\ldots .$