Key Terms

Key Terms

binomial series

the Maclaurin series for $f\left(x\right)={\left(1+x\right)}^r;$ it is given by
${\left(1+x\right)}^r={\displaystyle \sum _{n=0}^{\infty }\left(\begin{array}{c}r\hfill \\ n\hfill \end{array}\right)x^n}=1+rx+\frac{r\left(r-1\right)}{2\text{!}}{x}^2+\text{⋯}+\frac{r\left(r-1\right)\text{⋯}\left(r-n+1\right)}{n\text{!}}{x}^n+\text{⋯}$ 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 nth Taylor polynomial for $f$ at 0 is the nth 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 ${\displaystyle \sum _{n=0}^{\infty }{c}_n{x}^n}$ is a power series centered at $x=0\text{;}$ a series of the form ${\displaystyle \sum _{n=0}^{\infty }{c}_n{\left(x-a\right)}^n}$ 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 nth Taylor polynomial for $f$ at $x=a$ is $p_n\left(x\right)=f\left(a\right)+f^{\prime }\left(a\right)\left(x-a\right)+\frac{f^{″}\left(a\right)}{2\text{!}}{\left(x-a\right)}^2+\text{⋯}+\frac{f^{\left(n\right)}\left(a\right)}{n\text{!}}{\left(x-a\right)}^n$

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

term-by-term integration of a power series

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