Math 1220 Chapter 9 - Study Guide May 31, 2017 Sequences • Understand the formal definition of convergence of a sequence. Definition 9.1.2. • Discuss the convergence/divergence of a sequence. Example 9.1.4. • Determine if a sequence is increasing/decreasing and bounded. Example 9.1.8 & 9.1.9. Series • Discuss the convergence/divergence of a series. • Evaluate a telescoping series. Example 9.2.2. • Calculate the bounds on an approximation of a convergent series. Example 9.3.2. • Calculate the value of a geometric series. Example 9.2.4. • Express a repeating decimal in terms of a geometric series. Example 9.2.3. • Given a series, determine the appropriate test to ascertain convergence and correctly employ theorems. { Theorem 9.3.1. The Integral Test { Theorem 9.4.1. Direct Comparison Test { Theorem 9.4.2. Limit Comparison Test { Theorem 9.5.3. The Ratio Test { Theorem 9.5.4. The Root Test { Theorem 9.6.1. Alternating Series Test for Convergence 1 1 1 1 X ln n X 2n − 1 X en X 1 1. 2. 3. 4. n2 + 1 3n + 1 nn 3n − n + 1 n=1 n=1 n=1 n=1 Power Series • Identify the power series for functions like the following. 1 1 1 1. 2. 3. 1 − x 1 + x 1 − x2 • Determine the interval of convergence of a series. i.e. Example 9.7.5 & 9.7.6. • Find the power series of a function by differentiation and integration. 1. ln(1 + x) 2. arctan x Taylor and MacLaurin Series • Calculate the series for a given function. i.e. Example 9.8.4. & 9.8.6. • Use a Taylor or MacLaurin series to approximate an integral (just a second or third order approximation.) 2 1 1 Z Z 2 Z 1. ln x dx 2. e−x dx 3. arctan x dx 1 −1 0 Euler's Formula eix = cos x + i sin x. • Use Eulers Formula to prove trigonometric identities. Example 9.8.7 & 9.8.8. Binomial Theorem. • Use the Binomial Theorem to construct a power series of (1 ± αx)m. Example 9.10.2 & 9.10.3. Math 1220 , Page 2 of 2 The following list of theorems will be useful. Theorem 1 For all p 2 R, ln x < xp for sufficiently large x. 1 X Theorem 2 If lim an 6= 0: then an diverges. n!1 n=1 1 1 X X Theorem 3 If an and bn are convergent series, then so are the series n=1 n=1 1 1 1 1 1 1 1 1 X X X X X X X X c an = c an; (an + bn) = an + bn; and (an − bn) = an − bn: n=1 n=1 n=1 n=1 n=1 n=1 n=1 n=1 Theorem 4 The Integral Test: Suppose that fang is a sequence of positive terms, that f(x) is a continuous, positive, decreasing function of x for x ≥ N (N a positive integer), and that f(n) = an: 1 X Z 1 Then the series an and the integral f(x) dx either both converge or both diverge. n=N N 1 1 X X Theorem 5 Direct Comparison Test Suppose that an and bn are series with nonnegative terms. n=1 n=1 1 1 X X (a) If bn converges and for all n large enough, an ≤ bn, then an also converges. n=1 n=1 1 1 X X (b) If bn diverges and for all n large enough, an ≥ bn, then an also diverges. n=1 n=1 1 1 X X an Theorem 6 Limit Comparison Test Suppose that an and bn are series with nonnegative terms. If lim = c; n!1 b n=1 n=1 n 1 1 X X for some finite c > 0 , then an and bn both converge or both diverge. n=1 n=1 Theorem 7 The Ratio Test 8 1 ρ < 1; the series converges X an+1 < If an is a series with nonnegative terms, and lim = ρ. Then if ρ > 1; the series diverges n!1 a n=1 n : ρ = 1; the test provides NO information Theorem 8 The Root Test 8 1 ρ < 1; the series converges X pn < If an is a series with nonnegative terms, and lim janj = ρ. Then if ρ > 1; the series diverges n!1 n=1 : ρ = 1; the test provides NO information Theorem 9 Alternating Series Test for Convergence An alternating series converges if two simple conditions are met: (a) Terms are nonincreasing in absolute value, i.e. 0 ≤ un+1 ≤ un, for sufficiently large n. (b) lim un = 0. n!1.