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

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

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

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

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

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