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

∞ ∞ ∞ ∞ 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 ∈ R, ln x < xp for suﬃciently large x.

∞ X Theorem 2 If lim an 6= 0. then an diverges. n→∞ n=1

∞ ∞ X X Theorem 3 If an and bn are convergent series, then so are the series n=1 n=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 {an} 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. ∞ X Z ∞ Then the series an and the integral f(x) dx either both converge or both diverge. n=N N

∞ ∞ X X Theorem 5 Direct Comparison Test Suppose that an and bn are series with nonnegative terms. n=1 n=1

∞ ∞ X X (a) If bn converges and for all n large enough, an ≤ bn, then an also converges. n=1 n=1

∞ ∞ X X (b) If bn diverges and for all n large enough, an ≥ bn, then an also diverges. n=1 n=1

∞ ∞ X X an Theorem 6 Limit Comparison Test Suppose that an and bn are series with nonnegative terms. If lim = c, n→∞ b n=1 n=1 n ∞ ∞ 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  ∞ 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→∞ a n=1 n  ρ = 1, the test provides NO information

Theorem 8 The Root Test  ∞ ρ < 1, the series converges X pn  If an is a series with nonnegative terms, and lim |an| = ρ. Then if ρ > 1, the series diverges n→∞ 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→∞