Section 9.6 Calculus BC AP/Dual, Revised ©2018 viet.dang@humbleisd.net Ratio and Root Test Section 9.6 Calculus BC AP/Dual, Revised ©2018 viet.dang@humbleisd.net 12/18/2019 3:12 AM §9.6: Ratio and Root 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? 12/18/2019 3:12 AM §9.6: Ratio and Root Test

Ratio Test Let 𝒏=𝟏 ∞ 𝒂 𝒏 be a series in which 𝒂 𝒏 >𝟎 for all 𝒏 (or at least all 𝒏 past some particular threshold value 𝑵). Form the ratio 𝒂 𝒏+𝟏 𝒂 𝒏 and evaluate its limit as 𝒏→∞. Provided this limit exists, there are three possible cases: If 𝐥𝐢𝐦 𝒏→∞ 𝒂 𝒏+𝟏 𝒂 𝒏 <𝟏 , then 𝒏=𝟏 ∞ 𝒂 𝒏 converges If 𝐥𝐢𝐦 𝒏→∞ 𝒂 𝒏+𝟏 𝒂 𝒏 >𝟏 , then 𝒏=𝟏 ∞ 𝒂 𝒏 diverges If 𝐥𝐢𝐦 𝒏→∞ 𝒂 𝒏+𝟏 𝒂 𝒏 =𝟏 , the Ratio Test is inconclusive. Then, 𝒏=𝟏 ∞ 𝒂 𝒏 could either converge or diverge as another test is needed to decide the series. 12/18/2019 3:12 AM §9.6: Ratio and Root Test

Example 1 Determine whether the following converges or diverges of, 𝒏=𝟏 ∞ 𝟒 𝒏 𝒏! 12/18/2019 3:12 AM §9.6: Ratio and Root Test

Example 2 Determine whether the following converges or diverges of, 𝒏=𝟏 ∞ (−𝟏𝟎) 𝒏 𝟒 𝟐𝒏+𝟏 𝒏+𝟏 12/18/2019 3:12 AM §9.6: Ratio and Root Test

Example 2 Determine whether the following converges or diverges of, 𝒏=𝟏 ∞ (−𝟏𝟎) 𝒏 𝟒 𝟐𝒏+𝟏 𝒏+𝟏 12/18/2019 3:12 AM §9.6: Ratio and Root Test

Example 3 Determine whether the following converges or diverges of, 𝒏=𝟏 ∞ 𝒏! 𝟓 𝒏 12/18/2019 3:12 AM §9.6: Ratio and Root Test

Example 4 Determine whether the following converges or diverges of, 𝒏=𝟏 ∞ 𝒏+𝟐 𝟐𝒏+𝟕 12/18/2019 3:12 AM §9.6: Ratio and Root Test

Your Turn Determine whether the following converges or diverges of, 𝒏=𝟏 ∞ 𝒏 𝟐 (𝟑 𝒏+𝟏 ) 𝟐 𝒏 12/18/2019 3:12 AM §9.6: Ratio and Root Test

Root Test 𝒏=𝟏 ∞ 𝒂 𝒏 converges if 𝐥𝐢𝐦 𝒏→∞ 𝒏 𝒂 𝒏 <𝟏 𝒏=𝟏 ∞ 𝒂 𝒏 diverges if 𝐥𝐢𝐦 𝒏→∞ 𝒏 𝒂 𝒏 >𝟏 If 𝐥𝐢𝐦 𝒏→∞ 𝒏 𝒂 𝒏 =𝟏 , the Root Test is inconclusive so another test would be used. 12/18/2019 3:12 AM §9.6: Ratio and Root Test

Example 5 Determine whether the following converges or diverges of, 𝒏=𝟏 ∞ 𝒆 𝟐𝒏 𝒏 𝒏 12/18/2019 3:12 AM §9.6: Ratio and Root Test

Example 6 Determine whether the following converges or diverges of, 𝒏=𝟏 ∞ −𝟑𝒏 𝟐𝒏+𝟏 𝒏 12/18/2019 3:12 AM §9.6: Ratio and Root Test

Example 7 Determine whether the following converges or diverges of, 𝒏=𝟏 ∞ 𝒏 𝟑 𝟑 𝒏 12/18/2019 3:12 AM §9.6: Ratio and Root Test

Example 7* Determine whether the following converges or diverges of, 𝒏=𝟏 ∞ 𝒏 𝟑 𝟑 𝒏 12/18/2019 3:12 AM §9.6: Ratio and Root Test

Example 7* Determine whether the following converges or diverges of, 𝒏=𝟏 ∞ 𝒏 𝟑 𝟑 𝒏 12/18/2019 3:12 AM §9.6: Ratio and Root Test

Your Turn Determine whether if the root test converges or diverges of 𝒏=𝟏 ∞ 𝟑𝒏+𝟒 𝟐𝒏 𝒏 12/18/2019 3:12 AM §9.6: Ratio and Root Test

Assignment Page 633 13-31 odd, 35-47 EOO, 51-65 odd 12/18/2019 3:12 AM §9.6: Ratio and Root Test