### Key Terms

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

${\left(1+x\right)}^{r}={\displaystyle \sum _{n=0}^{\infty}\left(\begin{array}{c}r\hfill \\ n\hfill \end{array}\right){x}^{n}}=1+rx+\frac{r\left(r-1\right)}{2\text{!}}{x}^{2}+\text{\cdots}+\frac{r\left(r-1\right)\text{\cdots}\left(r-n+1\right)}{n\text{!}}{x}^{n}+\text{\cdots}$ for $\left|x\right|<1$

- interval of convergence
- the set of real numbers
*x*for which a power series converges

- Maclaurin polynomial
- a Taylor polynomial centered at 0; the
*n*th Taylor polynomial for $f$ at 0 is the*n*th Maclaurin polynomial for $f$

- Maclaurin series
- a Taylor series for a function $f$ at $x=0$ is known as a Maclaurin series for $f$

- nonelementary integral
- an integral for which the antiderivative of the integrand cannot be expressed as an elementary function

- power series
- a series of the form $\sum _{n=0}^{\infty}{c}_{n}{x}^{n}$ is a power series centered at $x=0\text{;}$ a series of the form $\sum _{n=0}^{\infty}{c}_{n}{\left(x-a\right)}^{n}$ is a power series centered at $x=a$

- radius of convergence
- if there exists a real number $R>0$ such that a power series centered at $x=a$ converges for $\left|x-a\right|<R$ and diverges for $\left|x-a\right|>R,$ then
*R*is the radius of convergence; if the power series only converges at $x=a,$ the radius of convergence is $R=0\text{;}$ if the power series converges for all real numbers*x*, the radius of convergence is $R=\infty $

- Taylor polynomials
- the
*n*th Taylor polynomial for $f$ at $x=a$ is ${p}_{n}\left(x\right)=f\left(a\right)+{f}^{\prime}\left(a\right)\left(x-a\right)+\frac{f\text{\u2033}\left(a\right)}{2\text{!}}{\left(x-a\right)}^{2}+\text{\cdots}+\frac{{f}^{\left(n\right)}\left(a\right)}{n\text{!}}{\left(x-a\right)}^{n}$

- Taylor series
- a power series at
*a*that converges to a function $f$ on some open interval containing*a*

- Taylor’s theorem with remainder
- for a function $f$ and the
*n*th Taylor polynomial for $f$ at $x=a,$ the remainder ${R}_{n}\left(x\right)=f\left(x\right)-{p}_{n}\left(x\right)$ satisfies ${R}_{n}\left(x\right)=\frac{{f}^{\left(n+1\right)}\left(c\right)}{\left(n+1\right)\text{!}}{\left(x-a\right)}^{n+1}$

for some*c*between*x*and*a*; if there exists an interval*I*containing*a*and a real number*M*such that $\left|{f}^{\left(n+1\right)}\left(x\right)\right|\le M$ for all*x*in*I*, then $\left|{R}_{n}\left(x\right)\right|\le \frac{M}{\left(n+1\right)\text{!}}{\left|x-a\right|}^{n+1}$

- term-by-term differentiation of a power series
- a technique for evaluating the derivative of a power series $\sum _{n=0}^{\infty}{c}_{n}{\left(x-a\right)}^{n}$ by evaluating the derivative of each term separately to create the new power series $\sum _{n=1}^{\infty}n{c}_{n}{\left(x-a\right)}^{n-1}$

- term-by-term integration of a power series
- a technique for integrating a power series $\sum _{n=0}^{\infty}{c}_{n}{\left(x-a\right)}^{n}$ by integrating each term separately to create the new power series $C+{\displaystyle \sum _{n=0}^{\infty}{c}_{n}\frac{{\left(x-a\right)}^{n+1}}{n+1}}$