Chapter 9 · Introduction

Chapter 9 — Power Series

Infinite polynomials — how a calculator evaluates $\sin x$, $e^x$ and $\ln x$ using nothing but addition and multiplication.

Press sin 0.3 on a calculator and the answer $0.2955202067$ appears instantly. But the chip inside can only add, subtract, multiply and divide — it has no little triangle to measure. So how does it know? The secret is that $\sin x$ can be rewritten as a polynomial that never ends: $$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\cdots$$ Three terms already give $0.3-\dfrac{0.027}{6}+\dfrac{0.00243}{120}=0.29552025$, correct to seven decimal places, using only $+$ and $\times$. The same trick works for $e^x$, $\ln(1+x)$, $\arctan x$ and many more functions.

An “infinite polynomial” $c_0+c_1x+c_2x^2+\cdots$ is called a power series. In Chapter 8 we learned when an infinite sum of numbers converges; now the terms depend on $x$, so the series may converge for some $x$ and diverge for others. This chapter finds out where a power series makes sense, how to build power series for familiar functions, and how to differentiate and integrate them as easily as polynomials. The payoff is enormous: approximations with guaranteed accuracy, limits and integrals that no other method can do (try $\int_0^1e^{-x^2}dx$), and the language engineers and physicists use every day.

Geometric series→Power series→Radius & interval of convergence→Differentiate & integrate term by term→Taylor & Maclaurin series→Approximations, limits & integrals

🎯 By the end of this chapter you can…

find the radius and interval of convergence of a power series, using the ratio test and checking the endpoints separately;
write functions such as $\dfrac{1}{1+x^2}$ or $\dfrac{x^3}{2-x}$ as power series by manipulating the geometric series;
differentiate and integrate power series term by term to obtain series for $\ln(1+x)$, $\arctan x$ and sums such as $\sum n x^n$;
build Taylor and Maclaurin series and Taylor polynomials, and bound the error of a Taylor approximation;
use Taylor series to approximate values, evaluate limits and integrals, and solve problems in science and engineering.

The dark curve is the true function; the blue curve is the sum of its first $n$ power-series terms. Watch the polynomial hug the curve over a wider and wider stretch as terms are added, and read off how accurate the value at the red point is. Every number in the readouts was computed with $+$ and $\times$ only.

Function:terms $n$:3point $x_0$:

$x_0$—

polynomial with $n$ terms—

true value—

error—

Notice two things in the teaser. Near $0$ a few terms are astonishingly accurate; far from $0$ you need many more. Understanding why — and how many terms are enough — is exactly what Sections 9.3 and 9.4 are about. First, Sections 9.1 and 9.2 build the general theory of power series, starting from the one series we already know how to sum: the geometric series.

Sections in this chapter

9.1

Power Series and Functions

The form $\sum c_n(x-a)^n$, radius and interval of convergence, and functions as geometric power series.

$\dfrac{1}{1-x}=\sum_{n=0}^{\infty}x^n,\ |x|\lt 1$

Open section →9.2

Properties of Power Series

Combine, differentiate and integrate term by term; series for $\ln(1+x)$ and $\arctan x$; uniqueness.

$\arctan x=\sum_{n=0}^{\infty}\dfrac{(-1)^nx^{2n+1}}{2n+1}$

Open section →9.3

Taylor and Maclaurin Series

Build the power series of a function from its derivatives; Taylor polynomials and their remainder.

$f(x)=\sum_{n=0}^{\infty}\dfrac{f^{(n)}(a)}{n!}(x-a)^n$

Open section →9.4

Working with Taylor Series

The binomial series, building new series from known ones, and series for limits, integrals and differential equations.

$e^x=\sum_{n=0}^{\infty}\dfrac{x^n}{n!}$

Open section →

✅ Before you start

This chapter rests on Chapter 8. Be sure you can sum a geometric series and know when it converges (8.2 Infinite Series), apply the ratio test (8.6 Ratio and Root Tests), and decide convergence at the endpoints with the $p$-series, comparison and alternating series tests (8.3, 8.4, 8.5). You will also differentiate and integrate polynomials constantly — review Calculus I 2.2 Differentiation Rules and 4.1 Antiderivatives if needed.

← 8.6 Ratio and Root Tests 9.1 Power Series and Functions →