23. Taylor Series

d.1. Taylor Polynomial Approximation

\(\underline{\sin x \quad \text{about } x=0}\)
\(\underline{\ln x \quad \text{about } x=1}\)
\(\underline{x^{1/3} \quad \text{about } x=8}\)

c. \(x^{1/3}\) about \(x=8\)

The Taylor series for \(x^{1/3}\) about \(x=8\) is \[\begin{aligned} x^{1/3}=&2+\dfrac{1}{12}(x-8)-\dfrac{1}{288}(x-8)^2+\dfrac{5}{20\,736}(x-8)^3 \\ &-\dfrac{5}{248\,832}(x-8)^4+\dfrac{11}{5\,971\,968}(x-8)^5+\cdots \end{aligned}\] Here are the graphs of \(x^{1/3}\) (in BLUE) with the \(0^\text{th}\) through \(8^\text{th}\) degree Taylor polynomial approximations (in RED):