Chapter 10: Infinite Sequences and Series
Sequences and series are applicable to many contexts, but of what use are series with infinitely many terms? The series \(1 + 2 + 3 + 4 + \cdots\) is clearly infinite, but what about the series \(1 + 1/2 + 1/4 + 1/8 + \cdots \ques\) Is this series also infinite, or do the terms decrease quickly enough for the sum to be finite? (Find out in Section 10.3.) In this chapter we will also learn to model complicated functions using polynomials involving their derivatives. Doing so enables us to evaluate integrals that don't have elementary antiderivatives, therefore expanding the possibilities of calculus.
Sections
10.1 Sequences
Overview of finite and infinite sequences. Determining convergence or divergence of infinite sequences. Writing sequences recursively and explicitly. Describing sequences as monotone, bounded, and increasing or decreasing, with Monotonic Sequence Theorem.10.2 Infinite Series and Divergence Test
Overview of infinite series. Use of Divergence Test to show that a series diverges. Defining the Harmonic series. Defining and calculating geometric and telescoping sums. Properties of infinite series.10.3 Integral Test
Use of Integral Test to test for convergence or divergence of a series, with discussion of conditions. Defining \(p\)-series. Using the Integral Test to construct error bounds.10.4 Comparison Tests
Limit Comparison Test and Direct Comparison Test. Determining whether a series converges or diverges by comparing it with another series whose convergence or divergence is known.10.5 Alternating Series
Introduction to alternating series. Testing whether an alternating series converges, with conditions. Discussion of Alternating Series Error Bound.10.6 Absolute Convergence and the Ratio and Root Tests
Definition of absolute convergence and conditional convergence. Testing for these types of convergences. Use of Ratio Test and Root Test to test for convergence or divergence of a series. Discussion of conditions.10.7 Power Series
Introduction to power series. Discussion of center and radius of convergence. Establishing interval of convergence and radius of convergence. Differentiating and integrating power series, including solving initial value problems. Manipulation of rational functions into power series. Maclaurin Series, Taylor Series, and Binomial Series. Use of Taylor's Remainder Theorem to prove a series' convergence and for error bounding.