Essential Concepts
Introduction to Power Series
- For a power series centered at [latex]x=a[/latex], one of the following three properties hold:
- The power series converges only at [latex]x=a[/latex]. In this case, we say that the radius of convergence is [latex]R=0[/latex].
- The power series converges for all real numbers x. In this case, we say that the radius of convergence is [latex]R=\infty[/latex].
- There is a real number R such that the series converges for [latex]|x-a|
- 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 [latex]\displaystyle\sum _{n=0}^{\infty }{x}^{n}=\frac{1}{1-x}[/latex] for [latex]|x|<1[/latex] allows us to represent certain functions using geometric series.
Operations with Power Series
- Given two power series [latex]\displaystyle\sum _{n=0}^{\infty }{c}_{n}{x}^{n}[/latex] and [latex]\displaystyle\sum _{n=0}^{\infty }{d}_{n}{x}^{n}[/latex] that converge to functions f and g on a common interval I, the sum and difference of the two series converge to [latex]f\pm g[/latex], respectively, on I. In addition, for any real number b and integer [latex]m\ge 0[/latex], the series [latex]\displaystyle\sum _{n=0}^{\infty }b{x}^{m}{c}_{n}{x}^{n}[/latex] converges to [latex]b{x}^{m}f\left(x\right)[/latex] and the series [latex]\displaystyle\sum _{n=0}^{\infty }{c}_{n}{\left(b{x}^{m}\right)}^{n}[/latex] converges to [latex]f\left(b{x}^{m}\right)[/latex] whenever bxm is in the interval I.
- Given two power series that converge on an interval [latex]\left(\text{-}R,R\right)[/latex], the Cauchy product of the two power series converges on the interval [latex]\left(\text{-}R,R\right)[/latex].
- Given a power series that converges to a function f on an interval [latex]\left(\text{-}R,R\right)[/latex], the series can be differentiated term-by-term and the resulting series converges to [latex]{f}^{\prime }[/latex] on [latex]\left(\text{-}R,R\right)[/latex]. The series can also be integrated term-by-term and the resulting series converges to [latex]\displaystyle\int f\left(x\right)dx[/latex] on [latex]\left(\text{-}R,R\right)[/latex].
Taylor and Maclaurin Series
- Taylor polynomials are used to approximate functions near a value [latex]x=a[/latex]. Maclaurin polynomials are Taylor polynomials at [latex]x=0[/latex].
- The nth degree Taylor polynomials for a function [latex]f[/latex] are the partial sums of the Taylor series for [latex]f[/latex].
- If a function [latex]f[/latex] has a power series representation at [latex]x=a[/latex], then it is given by its Taylor series at [latex]x=a[/latex].
- A Taylor series for [latex]f[/latex] converges to [latex]f[/latex] if and only if [latex]\underset{n\to \infty }{\text{lim}}{R}_{n}\left(x\right)=0[/latex] where [latex]{R}_{n}\left(x\right)=f\left(x\right)-{p}_{n}\left(x\right)[/latex].
- The Taylor series for ex, [latex]\sin{x}[/latex], and [latex]\cos{x}[/latex] converge to the respective functions for all real x.
Applications of 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.
Key Equations
- Power series centered at [latex]x=0[/latex]
[latex]\displaystyle\sum _{n=0}^{\infty }{c}_{n}{x}^{n}={c}_{0}+{c}_{1}x+{c}_{2}{x}^{2}+\cdots[/latex] - Power series centered at [latex]x=a[/latex]
[latex]\displaystyle\sum _{n=0}^{\infty }{c}_{n}{\left(x-a\right)}^{n}={c}_{0}+{c}_{1}\left(x-a\right)+{c}_{2}{\left(x-a\right)}^{2}+\cdots[/latex] - Taylor series for the function [latex]f[/latex] at the point [latex]x=a[/latex]
[latex]\displaystyle\sum _{n=0}^{\infty }\frac{{f}^{\left(n\right)}\left(a\right)}{n\text{!}}{\left(x-a\right)}^{n}=f\left(a\right)+{f}^{\prime }\left(a\right)\left(x-a\right)+\frac{f^{\prime\prime}\left(a\right)}{2\text{!}}{\left(x-a\right)}^{2}+\cdots +\frac{{f}^{\left(n\right)}\left(a\right)}{n\text{!}}{\left(x-a\right)}^{n}+\cdots[/latex]
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]
interval of convergence
the set of real numbers x for which a power series converges
- Maclaurin polynomial
- a Taylor polynomial centered at 0; the [latex]n[/latex]th Taylor polynomial for [latex]f[/latex] at 0 is the [latex]n[/latex]th Maclaurin polynomial for [latex]f[/latex]
- Maclaurin series
- a Taylor series for a function [latex]f[/latex] at [latex]x=0[/latex] is known as a Maclaurin series for [latex]f[/latex]
- 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 [latex]\displaystyle\sum _{n=0}^{\infty }{c}_{n}{x}^{n}[/latex] is a power series centered at [latex]x=0[/latex]; a series of the form [latex]\displaystyle\sum _{n=0}^{\infty }{c}_{n}{\left(x-a\right)}^{n}[/latex] is a power series centered at [latex]x=a[/latex]
- radius of convergence
- if there exists a real number [latex]R>0[/latex] such that a power series centered at [latex]x=a[/latex] converges for [latex]|x-a|
- Taylor polynomials
- the [latex]n[/latex]th Taylor polynomial for [latex]f[/latex] at [latex]x=a[/latex] is [latex]{p}_{n}\left(x\right)=f\left(a\right)+{f}^{\prime }\left(a\right)\left(x-a\right)+\frac{f^{\prime\prime}\left(a\right)}{2\text{!}}{\left(x-a\right)}^{2}+\cdots +\frac{{f}^{\left(n\right)}\left(a\right)}{n\text{!}}{\left(x-a\right)}^{n}[/latex]
- Taylor series
- a power series at [latex]a[/latex] that converges to a function [latex]f[/latex] on some open interval containing [latex]a[/latex]
- Taylor’s theorem with remainder
- for a function [latex]f[/latex] and the nth Taylor polynomial for [latex]f[/latex] at [latex]x=a[/latex], the remainder [latex]{R}_{n}\left(x\right)=f\left(x\right)-{p}_{n}\left(x\right)[/latex] satisfies [latex]{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}[/latex]
for some [latex]c[/latex] between [latex]x[/latex] and [latex]a[/latex]; if there exists an interval [latex]I[/latex] containing [latex]a[/latex] and a real number [latex]M[/latex] such that [latex]|{f}^{\left(n+1\right)}\left(x\right)|\le M[/latex] for all [latex]x[/latex] in [latex]I[/latex], then [latex]|{R}_{n}\left(x\right)|\le \frac{M}{\left(n+1\right)\text{!}}{|x-a|}^{n+1}[/latex]
term-by-term differentiation of a power series
a technique for evaluating the derivative of a power series [latex]\displaystyle\sum _{n=0}^{\infty }{c}_{n}{\left(x-a\right)}^{n}[/latex] by evaluating the derivative of each term separately to create the new power series [latex]\displaystyle\sum _{n=1}^{\infty }n{c}_{n}{\left(x-a\right)}^{n - 1}[/latex]
- term-by-term integration of a power series
- a technique for integrating a power series [latex]\displaystyle\sum _{n=0}^{\infty }{c}_{n}{\left(x-a\right)}^{n}[/latex] by integrating each term separately to create the new power series [latex]C+\displaystyle\sum _{n=0}^{\infty }{c}_{n}\frac{{\left(x-a\right)}^{n+1}}{n+1}[/latex]