Nth, Geometric, and Telescoping Test Section 9.2 Calculus BC AP/Dual, Revised ©2018 viet.dang@humbleisd.net 12/1/2019 11:15 AM §9.2: nth, Geometric, and Telescoping 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? 12/1/2019 11:15 AM §9.2: nth, Geometric, and Telescoping Test

§9.2: nth, Geometric, and Telescoping Test Definitions Series is the sum of the terms in a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely. A series can be written with summation symbol sigma, , the Greek letter “𝑺”. 𝑺𝒏 is often called an 𝒏𝒕𝒉 partial sum, since it can represent the sum of a certain ‘part’ of a sequence. Infinite series converges if the sequence of partial sums converges to some number, 𝑺𝒏. If the sequence of partial sums diverges, then the series diverges. 12/1/2019 11:15 AM §9.2: nth, Geometric, and Telescoping Test

What is Sigma Notation? "The summation from 𝟏 to ∞ of 𝟐𝒏+𝟏": Upper Limit Function Summation Lower Limit Known as “index” "The summation from 𝟏 to ∞ of 𝟐𝒏+𝟏": 12/1/2019 11:15 AM §9.2: nth, Geometric, and Telescoping Test

§9.2: nth, Geometric, and Telescoping Test Example 1 Given the series 𝒏=𝟏 ∞ 𝒏 𝟐 +𝟐 , find the first five terms of the sequence of partial sums, and list them below. Then, evaluate. Is the sequence of partial sums has a limit or bound? 12/1/2019 11:15 AM §9.2: nth, Geometric, and Telescoping Test

§9.2: nth, Geometric, and Telescoping Test Geometric Series Test Geometric series is in the form, 𝒏=𝟏 ∞ 𝒂 𝟏 ( 𝒓) 𝒏 or 𝒏=𝟎 ∞ 𝒂 𝟏 ( 𝒓) 𝒏−𝟏 ;𝒂≠𝟎 𝒂 𝟏 is the Initial term of the series 𝒓 is the common ratio Convergence vs. Divergence The geometric series converges if 𝒓 <𝟏 to the sum of 𝑺= 𝒂 𝟏 𝟏−𝒓 The geometric series diverges if 𝒓 ≥𝟏 12/1/2019 11:15 AM §9.2: nth, Geometric, and Telescoping Test

§9.2: nth, Geometric, and Telescoping Test Example 2 Determine whether the following series converge or diverge, 𝒏=𝟏 ∞ 𝟑 𝟐 𝒏 . If it converges, identify the sum. 12/1/2019 11:15 AM §9.2: nth, Geometric, and Telescoping Test

§9.2: nth, Geometric, and Telescoping Test Example 2 Determine whether the following series converge or diverge, 𝒏=𝟏 ∞ 𝟑 𝟐 𝒏 . If it converges, identify the sum. 12/1/2019 11:15 AM §9.2: nth, Geometric, and Telescoping Test

§9.2: nth, Geometric, and Telescoping Test Example 3 Determine whether the following series converge or diverge, 𝒏=𝟏 ∞ − 𝟑 𝟐 𝒏 12/1/2019 11:15 AM §9.2: nth, Geometric, and Telescoping Test

§9.2: nth, Geometric, and Telescoping Test Example 4 Determine whether the following series converge or diverge, 𝒏=𝟎 ∞ 𝟑 𝒏+𝟏 𝟓 𝒏 12/1/2019 11:15 AM §9.2: nth, Geometric, and Telescoping Test

§9.2: nth, Geometric, and Telescoping Test Example 4 Determine whether the following series converge or diverge, 𝒏=𝟎 ∞ 𝟑 𝒏+𝟏 𝟓 𝒏 12/1/2019 11:15 AM §9.2: nth, Geometric, and Telescoping Test

§9.2: nth, Geometric, and Telescoping Test Your Turn Determine whether the following series converge or diverge, 𝒏=𝟐 ∞ 𝟑 − 𝟏 𝟐 𝒏 . If it converges, identify the sum. 12/1/2019 11:15 AM §9.2: nth, Geometric, and Telescoping Test

Nth Term Test for Divergence One Way Test (Divergence Only) If 𝐥𝐢𝐦 𝒏→∞ 𝒂 𝒏 ≠𝟎 , then the series 𝒏=𝟏 ∞ 𝒂 𝒏 diverges Therefore, if 𝐥𝐢𝐦 𝒏→∞ 𝒂 𝒏 =𝟎 , then the series DOES NOT converge If the test does not pass, the test is INCONCLUSIVE and another test must be used Use this test FIRST before others, due to time constraints 12/1/2019 11:15 AM §9.2: nth, Geometric, and Telescoping Test

§9.2: nth, Geometric, and Telescoping Test Example 5 Determine whether the following series converge or diverge, 𝒏=𝟏 ∞ 𝟐𝒏+𝟑 𝟑𝒏−𝟓 . If it converges, identify the sum. 12/1/2019 11:15 AM §9.2: nth, Geometric, and Telescoping Test

§9.2: nth, Geometric, and Telescoping Test Example 6 Determine whether the following series converge or diverge, 𝒏=𝟏 ∞ 𝒏! 𝟐𝒏!+𝟏 . If it converges, identify the sum. 12/1/2019 11:15 AM §9.2: nth, Geometric, and Telescoping Test

§9.2: nth, Geometric, and Telescoping Test Example 7 Determine whether the following series converge or diverge, 𝒏=𝟐 ∞ 𝟏 𝟏.𝟏 𝒏 . 12/1/2019 11:15 AM §9.2: nth, Geometric, and Telescoping Test

§9.2: nth, Geometric, and Telescoping Test Example 7 Determine whether the following series converge or diverge, 𝒏=𝟐 ∞ 𝟏 𝟏.𝟏 𝒏 . 12/1/2019 11:15 AM §9.2: nth, Geometric, and Telescoping Test

§9.2: nth, Geometric, and Telescoping Test Your Turn Determine whether the following series converge or diverge, 𝒏=𝟏 ∞ 𝟑 𝒏 −𝟐 𝟑 𝒏 . If it converges, identify the sum. 12/1/2019 11:15 AM §9.2: nth, Geometric, and Telescoping Test

§9.2: nth, Geometric, and Telescoping Test Known as the terms “collapse” to one term or several terms Uses the associative property of addition Series collapses to a finite sum To get the sum, start plugging in numbers Hint: Generally has two ratios associated with Telescoping Series when generating the sum 12/1/2019 11:15 AM §9.2: nth, Geometric, and Telescoping Test

§9.2: nth, Geometric, and Telescoping Test Example 8 Determine whether the following series converge or diverge, 𝒏=𝟏 ∞ 𝟏 𝟐𝒏+𝟏 − 𝟏 𝟐𝒏+𝟑 𝒏=𝟏 ∞ 𝟏 𝟐𝒏+𝟏 − 𝟏 𝟐𝒏+𝟑 . If it converges, identify the sum. 12/1/2019 11:15 AM §9.2: nth, Geometric, and Telescoping Test

§9.2: nth, Geometric, and Telescoping Test Example 9 Determine whether the following series converge or diverge, 𝒏=𝟏 ∞ 𝟏 𝒏 𝟐 +𝟒𝒏+𝟑 . If it converges, identify the sum. 12/1/2019 11:15 AM §9.2: nth, Geometric, and Telescoping Test

§9.2: nth, Geometric, and Telescoping Test Example 9 Determine whether the following series converge or diverge, 𝒏=𝟏 ∞ 𝟏 𝒏 𝟐 +𝟒𝒏+𝟑 . If it converges, identify the sum. 12/1/2019 11:15 AM §9.2: nth, Geometric, and Telescoping Test

§9.2: nth, Geometric, and Telescoping Test Example 9 Determine whether the following series converge or diverge, 𝒏=𝟏 ∞ 𝟏 𝒏 𝟐 +𝟒𝒏+𝟑 . If it converges, identify the sum. 12/1/2019 11:15 AM §9.2: nth, Geometric, and Telescoping Test

§9.2: nth, Geometric, and Telescoping Test Example 9 Determine whether the following series converge or diverge, 𝒏=𝟏 ∞ 𝟏 𝒏 𝟐 +𝟒𝒏+𝟑 . If it converges, identify the sum. 12/1/2019 11:15 AM §9.2: nth, Geometric, and Telescoping Test

§9.2: nth, Geometric, and Telescoping Test Your Turn Determine whether the following series converge or diverge, 𝒏=𝟏 ∞ 𝟏 𝒏 𝒏+𝟏 . If it converges, identify the sum. 12/1/2019 11:15 AM §9.2: nth, Geometric, and Telescoping Test

AP Multiple Choice Practice Question 1 (non-calculator) What is the value of 𝒏=𝟏 ∞ 𝟐 𝒏+𝟏 𝟑 𝒏 ? (A) 𝟐 (B) 𝟒 (C) 𝟔 (D) The series diverges 12/1/2019 11:15 AM §9.2: nth, Geometric, and Telescoping Test

AP Multiple Choice Practice Question 1 (non-calculator) What is the value of 𝒏=𝟏 ∞ 𝟐 𝒏+𝟏 𝟑 𝒏 ? Vocabulary Connections and Process Answer 12/1/2019 11:15 AM §9.2: nth, Geometric, and Telescoping Test

§9.2: nth, Geometric, and Telescoping Test Assignment Page 601 7-19 odd, 25-31 odd, 41-53 odd 12/1/2019 11:15 AM §9.2: nth, Geometric, and Telescoping Test