Alternating Series Test Section 9.5 Calculus BC AP/Dual, Revised ©2018 viet.dang@humbleisd.net 10/24/2019 12:45 AM §9.5: Alternating Series Test

Summary of Tests for Series Looking at the first few terms of the sequence of partial sums may not help us much so we will learn the following ten tests to determine convergence or divergence: P 𝒑-series: Is the series in the form 𝟏 𝒏 𝑷 ? A Alternating series: Does the series alternate? If it does, are the terms getting smaller, and is the 𝒏th term 0? R Ratio Test: Does the series contain things that grow very large as 𝒏 increases (exponentials or factorials)? R Root Test: Does the series contain a radical? T Telescoping series: Will all but a couple of the terms in the series cancel out? I Integral Test: Can you easily integrate the expression that define the series? N 𝒏th Term divergence Test: Is the nth term something other than zero? G Geometric series: Is the series of the form, 𝒏=𝟎 ∞ 𝒂 𝒓 𝒏 C Comparison Tests: Is the series almost another kind of series (e.g. 𝒑-series or geometric)? Which would be better to use: Direct or Limit Comparison Test? 10/24/2019 12:45 AM §9.5: Alternating Series Test

Alternating Series Test 10/24/2019 12:45 AM §9.5: Alternating Series Test

Alternating Series Test Sum 10/24/2019 12:45 AM §9.5: Alternating Series Test

Alternating Series Test An alternating series is a series whose terms are alternatively positive and negative on consecutive terms (no two consecutive terms have the same sign) If 𝒂 𝒏 >𝟎, then the alternating series 𝒏=𝟏 ∞ −𝟏 𝒏 𝒂 𝒏 and 𝒏=𝟏 ∞ −𝟏 𝒏+𝟏 𝒂 𝒏 converges if the following conditions are met: Alternates in signs Decreases in magnitude 𝒂 𝒏+𝟏 < 𝒂 𝒏 for all 𝒏 Have a limit of zero 𝐥𝐢𝐦 𝒏→∞ 𝒂 𝒏 =𝟎 This does not say if 𝐥𝐢𝐦 𝒏→∞ 𝒂 𝒏 ≠𝟎, the series diverges by the AST. The AST CAN only be used to prove convergence. If 𝐥𝐢𝐦 𝒏→∞ 𝒂 𝒏 ≠𝟎, then the series diverges by the nth term test for divergence and not by the AST. Alternate Harmonic Series is NOT like the harmonic series. It will allow the series to converge to the sum of zero. 10/24/2019 12:45 AM §9.5: Alternating Series Test

Alternating Series Test Examples

Alternating Series Test 10/24/2019 12:45 AM §9.5: Alternating Series Test

§9.5: Alternating Series Test Example 1 Use the Alternating Series Test to prove whether the series converges or diverges, 𝒏=𝟏 ∞ −𝟏 𝒏 𝒏 𝟐 10/24/2019 12:45 AM §9.5: Alternating Series Test

§9.5: Alternating Series Test Example 1 Use the Alternating Series Test to prove whether the series converges or diverges, 𝒏=𝟏 ∞ −𝟏 𝒏 𝒏 𝟐 10/24/2019 12:45 AM §9.5: Alternating Series Test

§9.5: Alternating Series Test Example 2 Prove whether the series converges or diverges, 𝒏=𝟏 ∞ −𝟏 𝒏 𝟏 𝐥𝐧 𝒏+𝟏 10/24/2019 12:45 AM §9.5: Alternating Series Test

§9.5: Alternating Series Test Example 3 Prove whether the series converges or diverges, 𝒏=𝟏 ∞ −𝟏 𝒏+𝟏 𝒏 𝟐 𝒏 𝟐 +𝟓 10/24/2019 12:45 AM §9.5: Alternating Series Test

§9.5: Alternating Series Test Example 4 Prove whether the series converges or diverges, 𝒏=𝟏 ∞ 𝐜𝐨𝐬 𝒏𝝅 𝒏 10/24/2019 12:45 AM §9.5: Alternating Series Test

§9.5: Alternating Series Test Your Turn Prove whether the series converges or diverges, 𝒏=𝟏 ∞ −𝟏 𝒏−𝟏 𝒏! (Use the comparison for decreasing) 10/24/2019 12:45 AM §9.5: Alternating Series Test

Alternating Harmonic Series Harmonic Series is where the sum takes a long time before approaching zero and therefore, the terms of the series diverges. Alternating Harmonic Series is like the harmonic series where the terms eventually (and slowly approach) where the terms of the series approach zero. The terms of the series converges. 10/24/2019 12:45 AM §9.5: Alternating Series Test

§9.5: Alternating Series Test Example 5 Use the Alternating Series Test to prove whether the series converges or diverges, 𝒏=𝟏 ∞ −𝟏 𝒏 𝟏 𝒏 10/24/2019 12:45 AM §9.5: Alternating Series Test

§9.5: Alternating Series Test Example 5 Use the Alternating Series Test to prove whether the series converges or diverges, 𝒏=𝟏 ∞ −𝟏 𝒏 𝟏 𝒏 10/24/2019 12:45 AM §9.5: Alternating Series Test

§9.5: Alternating Series Test Your Turn Use the Alternating Series Test to prove whether the series converges or diverges, 𝒏=𝟏 ∞ −𝟏 𝒏+𝟏 𝟏 𝒏 10/24/2019 12:45 AM §9.5: Alternating Series Test

§9.5: Alternating Series Test Your Turn Use the Alternating Series Test to prove whether the series converges or diverges, 𝒏=𝟏 ∞ −𝟏 𝒏+𝟏 𝟏 𝒏 10/24/2019 12:45 AM §9.5: Alternating Series Test

Absolute vs Conditional Convergence Absolute Convergent is where the series of 𝒏=𝟏 ∞ 𝒂 𝒏 converges, then 𝒏=𝟏 ∞ 𝒂 𝒏 also converges. Conditionally Convergent is where the series of 𝒏=𝟏 ∞ 𝒂 𝒏 converges, then 𝒏=𝟏 ∞ 𝒂 𝒏 diverges Sometimes, a rearrangement of terms in a convergent alternating series can result in a different sum. 10/24/2019 12:45 AM §9.5: Alternating Series Test

§9.5: Alternating Series Test Absolute Convergence In absolutely convergent series, the terms are heading to terms of zero so quickly that sliding the terms around does not make any difference. In other words, addition in the absolutely convergent case follows the same rules of addition that we have always believed and treasured; addition is commutative for absolutely convergent series. Every rearrangement can be rearranged to have any sum. 10/24/2019 12:45 AM §9.5: Alternating Series Test

Conditional Convergence In a conditionally convergent series, the terms can be rearranged to have any sum. 10/24/2019 12:45 AM §9.5: Alternating Series Test

§9.5: Alternating Series Test Example 6 Determine whether 𝒏=𝟏 ∞ (−𝟏) 𝒏 𝒏 alternating series converges or diverges. If it converges, determine whether it is absolutely convergent or conditionally convergent. 10/24/2019 12:45 AM §9.5: Alternating Series Test

§9.5: Alternating Series Test Example 6 Determine whether 𝒏=𝟏 ∞ (−𝟏) 𝒏 𝒏 alternating series converges or diverges. If it converges, determine whether it is absolutely convergent or conditionally convergent. 10/24/2019 12:45 AM §9.5: Alternating Series Test

§9.5: Alternating Series Test Example 7 Determine whether 𝒏=𝟏 ∞ (−𝟏) 𝒏+𝟏 𝟑 𝒏 alternating series converges or diverges. If it converges, determine whether it is absolutely convergent or conditionally convergent. 10/24/2019 12:45 AM §9.5: Alternating Series Test

§9.5: Alternating Series Test Example 7 Determine whether 𝒏=𝟏 ∞ (−𝟏) 𝒏+𝟏 𝟑 𝒏 alternating series converges or diverges. If it converges, determine whether it is absolutely convergent or conditionally convergent. 10/24/2019 12:45 AM §9.5: Alternating Series Test

§9.5: Alternating Series Test Your Turn Determine whether 𝒏=𝟏 ∞ (−𝟏) 𝒏+𝟐 𝒏 𝟐 alternating series converges or diverges. If it converges, determine whether it is absolutely convergent or conditionally convergent. 10/24/2019 12:45 AM §9.5: Alternating Series Test

AP Multiple Choice Practice Question 1 (non-calculator) Which of the following series converge? I. 𝒏=𝟏 ∞ 𝟏 𝒏 𝟐 II. 𝒏=𝟏 ∞ 𝟏 𝒏 III. 𝒏=𝟏 ∞ (−𝟏) 𝒏 𝒏 (A) I only (B) II only (C) III only (D) I and III only 10/24/2019 12:45 AM §9.5: Alternating Series Test

AP Multiple Choice Practice Question 1 (non-calculator) Which of the following series converge? I. 𝒏=𝟏 ∞ 𝟏 𝒏 𝟐 , II. 𝒏=𝟏 ∞ 𝟏 𝒏 , III. 𝒏=𝟏 ∞ (−𝟏) 𝒏 𝒏 Vocabulary Connections and Process Answer 10/24/2019 12:45 AM §9.5: Alternating Series Test

§9.5: Alternating Series Test Assignment Page 625 5-25 EOO, 27, 29, 37-51 odd 10/24/2019 12:45 AM §9.5: Alternating Series Test