Summary of Working with Taylor Series

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.

Glossary

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