23. Taylor Series

d.1. Taylor Polynomial Approximation

b. \(\ln x\) about \(x=1\)

The Taylor series for \(\ln x\) about \(x=1\) is: \[\begin{aligned} \ln x=&(x-1)-\dfrac{(x-1)^2}{2}+\dfrac{(x-1)^3}{3} \\ &-\dfrac{(x-1)^4}{4}+\dfrac{(x-1)^5}{5}-\dfrac{(x-1)^6}{6}+\cdots \end{aligned}\] Here are the graphs of \(\ln x\) (in BLUE) with the \(1^\text{st}\) through \(10^\text{th}\) degree Taylor polynomial approximations (in RED):

The interval of convergence of this Taylor series is \((0,2]\). It is obvious from the plots that the series does not converge for \(x \gt 2\) and the function \(\ln x\) is not even defined for \(x \lt 0\).