Chapter Comments

• One goal of this chapter is to be able to approximate certain kinds of functions with polynomials. Recall from Calc I, specifically §3.10, that we could approximate a function f(x) at a point a with the tangent line to f at a:
  f(x)  f(a) + f'(a) (x-a)

  But most functions are curvy and the tangent line does not account for the "curvy-ness". So we would like better approximations. It actually turns out that including higher order derivatives in the approximating polynomial can do a pretty good job.

  Although the first part of Chapter 12 may seem unrelated to the goal of approximating certain functions with polynomials, this initial material (through section §12.7) will set the foundation.

  Another skill that you will practice in this chapter is that of reading mathematics. This chapter is notation-intensive, and to become comfortable with the content you will need to become comfortable interpreting the symbols. When you were learning to read and write you needed to understand the words you were reading and writing, and the letters that represented the words. You needed to be able to find and construct meaning in sentences -- which are combinations of words that follow certain patterns and rules. You want to practice reading and writing mathematical notation so that eventually (persist!!) you can use mathematical symbols as words and sentences. This is not an easy task and this chapter will challenge you to get better at it.

Chapter Contents

• §12.1 Sequences

• §12.2 Series

• §12.3 The Integral Test and Estimates of Sums

• §12.4 The Comparison Tests

• §12.5 Alternating Series

• §12.6 Absolute Convergence and the Ratio and Root Tests

• §12.7 Strategy for Testing Series

• §12.8 Power Series

• §12.9 Representations of Functions as Power Series

• §12.10, §12.11, §12.12 Taylor and Maclaurin Series, The Binomial Series, Applications of Taylor Polynomials

Chapter 12 Sample Test Items and Sample Solutions

This document is part of the Math 2433: Calculus and Analytic Geometry III course designed for the Independent Study Department in the College of Continuing Education at the University of Oklahoma, Norman.

Go to:

• Calculus III Syllabus

• Table of Contents for Chapter 11: Parametric Equations and Polar Coordinates
• Table of Contents for Chapter 13: Vectors and the Geometry of Space
• Table of Contents for Chapter 14: Vector Functions