## Summary of Working with Taylor Series

### Essential Concepts

• The binomial series is the Maclaurin series for $f\left(x\right)={\left(1+x\right)}^{r}$. It converges for $|x|<1$.
• 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 $f\left(x\right)={\left(1+x\right)}^{r}$; it is given by

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