Summary of Working with Taylor Series

Essential Concepts

  • The binomial series is the Maclaurin series for [latex]f\left(x\right)={\left(1+x\right)}^{r}[/latex]. It converges for [latex]|x|<1[/latex].
  • Taylor series for functions can often be derived by algebraic operations with a known Taylor series or by differentiating or integrating a known Taylor series.
  • Power series can be used to solve differential equations.
  • Taylor series can be used to help approximate integrals that cannot be evaluated by other means.


binomial series
the Maclaurin series for [latex]f\left(x\right)={\left(1+x\right)}^{r}[/latex]; it is given by

[latex]{\left(1+x\right)}^{r}=\displaystyle\sum _{n=0}^{\infty }\left(\begin{array}{c}r\hfill \\ n\hfill \end{array}\right){x}^{n}=1+rx+\frac{r\left(r - 1\right)}{2\text{!}}{x}^{2}+\cdots +\frac{r\left(r - 1\right)\cdots \left(r-n+1\right)}{n\text{!}}{x}^{n}+\cdots [/latex] for [latex]|x|<1[/latex]
nonelementary integral
an integral for which the antiderivative of the integrand cannot be expressed as an elementary function