Direct and Limit Comparison Test Section 9.4 Calculus BC AP/Dual, Revised ©2017 viet.dang@humbleisd.net 11/24/2018 5:15 PM §9.4: Direct and Limit Comparison Tests

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? 11/24/2018 5:15 PM §9.4: Direct and Limit Comparison Tests

Direct Comparison Test If 𝒂 𝒏 ≥𝟎 and 𝒃 𝒏 ≥𝟎 for all 𝑛 and 𝒂 𝒏 ≤ 𝒃 𝒏 then: If 𝒏=𝟏 ∞ 𝒃 𝒏 CONVERGES and 𝟎 ≤𝒂 𝒏 ≤ 𝒃 𝒏 , then 𝒏=𝟏 ∞ 𝒂 𝒏 CONVERGES as well. If 𝒏=𝟏 ∞ 𝒂 𝒏 DIVERGES and 𝟎 ≤𝒂 𝒏 ≤ 𝒃 𝒏 , then 𝒏=𝟏 ∞ 𝒃 𝒏 DIVERGES as well. In other words, smaller series is convergent if the larger series is convergent. In order to apply this test, both series need to start at the same 𝒌. The Direct Comparison Test works hand in hand with other tests. 11/24/2018 5:15 PM §9.4: Direct and Limit Comparison Tests

Direct Comparison Test If 𝒏=𝟏 ∞ 𝒃 𝒏 (given) CONVERGES and 𝟎≤ 𝒂 𝒏 ≤ 𝒃 𝒏 , then 𝒏=𝟏 ∞ 𝒂 𝒏 (parent function) CONVERGES too. 11/24/2018 5:15 PM §9.4: Direct and Limit Comparison Tests

Direct Comparison Test If 𝒏=𝟏 ∞ 𝒂 𝒏 (parent function) DIVERGES and 𝟎≤ 𝒂 𝒏 ≤ 𝒃 𝒏 , then 𝒏=𝟏 ∞ 𝒃 𝒏 (given) DIVERGES too. 11/24/2018 5:15 PM §9.4: Direct and Limit Comparison Tests

Steps for Direct Comparison Test Compare the given using the 𝒑-Series or GST (Test goes hand in hand) Start directly comparing by writing an inequality from the given to the end When comparing, drop off the constant and use the inequality Include the inequality when stating the conclusion of the test. Pick a number to correspond to 𝒏 to prove the test Identify whether the series converges or diverges 11/24/2018 5:15 PM §9.4: Direct and Limit Comparison Tests

§9.4: Direct and Limit Comparison Tests Example 1 Use the Direct Comparison Test to prove whether the series converges or diverges, 𝒏=𝟏 ∞ 𝟏 𝒏 𝟑 +𝟏 . 11/24/2018 5:15 PM §9.4: Direct and Limit Comparison Tests

§9.4: Direct and Limit Comparison Tests Example 1 Use the Direct Comparison Test to prove whether the series converges or diverges, 𝒏=𝟏 ∞ 𝟏 𝒏 𝟑 +𝟏 . 11/24/2018 5:15 PM §9.4: Direct and Limit Comparison Tests

§9.4: Direct and Limit Comparison Tests Example 2 Use the Direct Comparison Test to prove whether the series converges or diverges, 𝒏=𝟏 ∞ 𝟏 𝟑 𝒏 +𝟐 . 11/24/2018 5:15 PM §9.4: Direct and Limit Comparison Tests

§9.4: Direct and Limit Comparison Tests Example 2 Use the Direct Comparison Test to prove whether the series converges or diverges, 𝒏=𝟏 ∞ 𝟏 𝟑 𝒏 +𝟐 . 11/24/2018 5:15 PM §9.4: Direct and Limit Comparison Tests

§9.4: Direct and Limit Comparison Tests Your Turn Use the Direct Comparison Test to prove whether the series converges or diverges, 𝒏=𝟏 ∞ 𝟑 𝒏 𝟕 𝒏 +𝟏 . 11/24/2018 5:15 PM §9.4: Direct and Limit Comparison Tests

§9.4: Direct and Limit Comparison Tests Your Turn Use the Direct Comparison Test to prove whether the series converges or diverges, 𝒏=𝟏 ∞ 𝟑 𝒏 𝟕 𝒏 +𝟏 . 11/24/2018 5:15 PM §9.4: Direct and Limit Comparison Tests

§9.4: Direct and Limit Comparison Tests Example 3 Use the Direct Comparison Test to prove whether the series converges or diverges, 𝒏=𝟒 ∞ 𝟏 𝒏 −𝟏 . 11/24/2018 5:15 PM §9.4: Direct and Limit Comparison Tests

§9.4: Direct and Limit Comparison Tests Example 3 Use the Direct Comparison Test to prove whether the series converges or diverges, 𝒏=𝟒 ∞ 𝟏 𝒏 −𝟏 . 11/24/2018 5:15 PM §9.4: Direct and Limit Comparison Tests

§9.4: Direct and Limit Comparison Tests Your Turn Use the Direct Comparison Test to prove whether the series converges or diverges, 𝒏=𝟐 ∞ 𝒏 𝒏 𝟐 −𝟏 . 11/24/2018 5:15 PM §9.4: Direct and Limit Comparison Tests

§9.4: Direct and Limit Comparison Tests Your Turn Use the Direct Comparison Test to prove whether the series converges or diverges, 𝒏=𝟐 ∞ 𝒏 𝒏 𝟐 −𝟏 . 11/24/2018 5:15 PM §9.4: Direct and Limit Comparison Tests

§9.4: Direct and Limit Comparison Tests Review Example Prove the series converges for 𝒏=𝟏 ∞ 𝒏 𝟒 +𝟏𝟎 𝟒𝒏 𝟓 − 𝒏 𝟑 +𝟕 . 11/24/2018 5:15 PM §9.4: Direct and Limit Comparison Tests

§9.4: Direct and Limit Comparison Tests Review Example Prove the series converges for 𝒏=𝟏 ∞ 𝒏 𝟒 +𝟏𝟎 𝟒𝒏 𝟓 − 𝒏 𝟑 +𝟕 . 11/24/2018 5:15 PM §9.4: Direct and Limit Comparison Tests

§9.4: Direct and Limit Comparison Tests If 𝒂 𝒏 >𝟎 and 𝒃 𝒏 >𝟎 AND 𝐥𝐢𝐦 𝒏→∞ 𝒂 𝒏 𝒃 𝒏 =𝑳 or 𝐥𝐢𝐦 𝒏→∞ 𝒃 𝒏 𝒂 𝒏 =𝑳 where 𝑳 is both finite and positive: Then, 𝒏=𝟏 ∞ 𝒂 𝒏 and 𝒏=𝟏 ∞ 𝒃 𝒏 both CONVERGES or both DIVERGES Sometimes, the inequalities do not work or are difficult to show but the answer is given, this test will help prove it The reasoning behind this test is to CONFIRM whether the series converges or diverges (we don’t know what they are) as opposed to DCT, where we know what converges or diverges 11/24/2018 5:15 PM §9.4: Direct and Limit Comparison Tests

Steps for Limit Comparison Test Compare the given using the 𝒑-Series or GST Compare the original problem and divide it with the new answer When comparing, drop off the constant and use the inequality Drop all constants with the series equation Solve the limit using B/S, S/B, Same LC, or L’Hopital rules Determine whether the answer prove whether it is FINITE and POSITIVE 11/24/2018 5:15 PM §9.4: Direct and Limit Comparison Tests

§9.4: Direct and Limit Comparison Tests Example 4 Use the Limit Comparison test to prove whether the series converges or diverges, 𝒏=𝟏 ∞ 𝒏 𝟒 +𝟏𝟎 𝟒𝒏 𝟓 − 𝒏 𝟑 +𝟕 . 11/24/2018 5:15 PM §9.4: Direct and Limit Comparison Tests

§9.4: Direct and Limit Comparison Tests Example 4 Use the Limit Comparison test to prove whether the series converges or diverges, 𝒏=𝟏 ∞ 𝒏 𝟒 +𝟏𝟎 𝟒𝒏 𝟓 − 𝒏 𝟑 +𝟕 . 11/24/2018 5:15 PM §9.4: Direct and Limit Comparison Tests

Example 4 Use the Limit Comparison test to prove whether the series converges or diverges, 𝒏=𝟏 ∞ 𝒏 𝟒 +𝟏𝟎 𝟒𝒏 𝟓 − 𝒏 𝟑 +𝟕 . The answer is positive and finite. 11/24/2018 5:15 PM §9.4: Direct and Limit Comparison Tests

§9.4: Direct and Limit Comparison Tests Example 5 Use the Limit Comparison test to prove whether the series converges or diverges, 𝒏=𝟏 ∞ 𝒏 𝒏 𝟐 +𝟏 . 11/24/2018 5:15 PM §9.4: Direct and Limit Comparison Tests

§9.4: Direct and Limit Comparison Tests Example 5 Use the Limit Comparison test to prove whether the series converges or diverges, 𝒏=𝟏 ∞ 𝒏 𝒏 𝟐 +𝟏 . 11/24/2018 5:15 PM §9.4: Direct and Limit Comparison Tests

Example 5 Use the Limit Comparison test to prove whether the series converges or diverges, 𝒏=𝟏 ∞ 𝒏 𝒏 𝟐 +𝟏 . The answer is positive and finite. 11/24/2018 5:15 PM §9.4: Direct and Limit Comparison Tests

§9.4: Direct and Limit Comparison Tests Example 6 Use the Limit Comparison test to prove whether the series converges or diverges, 𝒏=𝟏 ∞ 𝟐 𝒏 𝒏 𝒏 𝟐 −𝟑 . 11/24/2018 5:15 PM §9.4: Direct and Limit Comparison Tests

§9.4: Direct and Limit Comparison Tests Example 6 Use the Limit Comparison test to prove whether the series converges or diverges, 𝒏=𝟏 ∞ 𝟐 𝒏 𝒏 𝒏 𝟐 −𝟑 . 11/24/2018 5:15 PM §9.4: Direct and Limit Comparison Tests

Example 6 Use the Limit Comparison test to prove whether the series converges or diverges, 𝒏=𝟏 ∞ 𝟐 𝒏 𝒏 𝒏 𝟐 −𝟑 . The answer is positive and finite. 11/24/2018 5:15 PM §9.4: Direct and Limit Comparison Tests

§9.4: Direct and Limit Comparison Tests Your Turn Use the Limit Comparison test to prove whether the series converges or diverges, 𝒏=𝟏 ∞ 𝟓𝒏−𝟑 𝒏 𝟑 −𝟐𝒏+𝟓 . 11/24/2018 5:15 PM §9.4: Direct and Limit Comparison Tests

§9.4: Direct and Limit Comparison Tests Your Turn Use the Limit Comparison test to prove whether the series converges or diverges, 𝒏=𝟏 ∞ 𝟓𝒏−𝟑 𝒏 𝟑 −𝟐𝒏+𝟓 . 11/24/2018 5:15 PM §9.4: Direct and Limit Comparison Tests

AP Multiple Choice Practice Question 1 (non-calculator) If 𝒏=𝟏 ∞ 𝒂 𝒏 diverges and 𝟎≤ 𝒂 𝒏 ≤ 𝒃 𝒏 for all 𝒏, which of the following statements must be true? (A) 𝒏=𝟏 ∞ −𝟏 𝒏 𝒂 𝒏 converges. (B) 𝒏=𝟏 ∞ −𝟏 𝒏 𝒃 𝒏 diverges. (C) 𝒏=𝟏 ∞ 𝒃 𝒏 converges. (D) 𝒏=𝟏 ∞ 𝒃 𝒏 diverges. 11/24/2018 5:15 PM §9.4: Direct and Limit Comparison Tests

AP Multiple Choice Practice Question 1 (non-calculator) If 𝒏=𝟏 ∞ 𝒂 𝒏 diverges and 𝟎≤ 𝒂 𝒏 ≤ 𝒃 𝒏 , for all 𝒏, which of the following statements must be true? Vocabulary Connections and Process Answer If 𝒏=𝟏 ∞ 𝒂 𝒏 (parent function) DIVERGES and 𝟎≤ 𝒂 𝒏 ≤ 𝒃 𝒏 , then 𝒏=𝟏 ∞ 𝒃 𝒏 (given) DIVERGES too. 11/24/2018 5:15 PM §9.4: Direct and Limit Comparison Tests

AP Multiple Choice Practice Question 2 (non-calculator) What series should we use in the limit comparison test in order to determine whether 𝑺= 𝒏=𝟏 ∞ 𝟐 𝒏 𝟑 𝒏 −𝟏 converges? (A) 𝒏=𝟏 ∞ 𝟑 𝒏 (B) 𝒏=𝟏 ∞ 𝟏 𝟑 𝒏 −𝟏 (C) 𝒏=𝟏 ∞ 𝟐 𝟑 𝒏 (D) 𝒏=𝟏 ∞ 𝟐 𝒏 11/24/2018 5:15 PM §9.4: Direct and Limit Comparison Tests

AP Multiple Choice Practice Question 2 (non-calculator) What series should we use in the limit comparison test in order to determine whether 𝑺= 𝒏=𝟏 ∞ 𝟐 𝒏 𝟑 𝒏 −𝟏 converges? Vocabulary Connections and Process Answer 11/24/2018 5:15 PM §9.4: Direct and Limit Comparison Tests

§9.4: Direct and Limit Comparison Tests Assignment Page 616 3-23 odd, 23-30 all, 33-41 odd 11/24/2018 5:15 PM §9.4: Direct and Limit Comparison Tests