### Key Concepts

## 6.1 Power Series and Functions

- For a power series centered at $x=a,$ one of the following three properties hold:
- The power series converges only at $x=a.$ In this case, we say that the radius of convergence is $R=0.$
- The power series converges for all real numbers
*x*. In this case, we say that the radius of convergence is $R=\infty .$ - There is a real number
*R*such that the series converges for $\left|x-a\right|<R$ and diverges for $\left|x-a\right|>R.$ In this case, the radius of convergence is*R*.

- If a power series converges on a finite interval, the series may or may not converge at the endpoints.
- The ratio test may often be used to determine the radius of convergence.
- The geometric series $\sum _{n=0}^{\infty}{x}^{n}}=\frac{1}{1-x$ for $\left|x\right|<1$ allows us to represent certain functions using geometric series.

## 6.2 Properties of Power Series

- Given two power series $\sum _{n=0}^{\infty}{c}_{n}{x}^{n}$ and $\sum _{n=0}^{\infty}{d}_{n}{x}^{n}$ that converge to functions
*f*and*g*on a common interval*I*, the sum and difference of the two series converge to $f\pm g,$ respectively, on*I*. In addition, for any real number*b*and integer $m\ge 0,$ the series $\sum _{n=0}^{\infty}b{x}^{m}{c}_{n}{x}^{n}$ converges to $b{x}^{m}f\left(x\right)$ and the series $\sum _{n=0}^{\infty}{c}_{n}{\left(b{x}^{m}\right)}^{n}$ converges to $f\left(b{x}^{m}\right)$ whenever*bx*is in the interval^{m}*I*. - Given two power series that converge on an interval $\left(\text{\u2212}R,R\right),$ the Cauchy product of the two power series converges on the interval $\left(\text{\u2212}R,R\right).$
- Given a power series that converges to a function
*f*on an interval $\left(\text{\u2212}R,R\right),$ the series can be differentiated term-by-term and the resulting series converges to ${f}^{\prime}$ on $\left(\text{\u2212}R,R\right).$ The series can also be integrated term-by-term and the resulting series converges to $\int f\left(x\right)\phantom{\rule{0.1em}{0ex}}dx$ on $\left(\text{\u2212}R,R\right).$

## 6.3 Taylor and Maclaurin Series

- Taylor polynomials are used to approximate functions near a value $x=a.$ Maclaurin polynomials are Taylor polynomials at $x=0.$
- The
*n*th degree Taylor polynomials for a function $f$ are the partial sums of the Taylor series for $f.$ - If a function $f$ has a power series representation at $x=a,$ then it is given by its Taylor series at $x=a.$
- A Taylor series for $f$ converges to $f$ if and only if $\underset{n\to \infty}{\text{lim}}{R}_{n}\left(x\right)=0$ where ${R}_{n}\left(x\right)=f\left(x\right)-{p}_{n}\left(x\right).$
- The Taylor series for
*e*, $\text{sin}\phantom{\rule{0.1em}{0ex}}x,$ and $\text{cos}\phantom{\rule{0.1em}{0ex}}x$ converge to the respective functions for all real^{x}*x*.

## 6.4 Working with Taylor Series

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