Worksheet-Section 5.6 Ratio and Root Test
.pdf
School
Georgia State University**We aren't endorsed by this school
Course
HIST 2110
Subject
Mathematics
Date
Dec 17, 2024
Pages
3
Uploaded by JusticeBook34240
Math 2212 – Calculus of One Variable II Worksheet 5.6 15.6 Ratio and Root Tests Learning Objectives 1.Use the ratio test to determine absolute convergence of a series. 2.Use the root test to determine absolute convergence of a series. 3.Describe a strategy for testing the convergence of a given series. The Ratio Test: Let ∑𝑎𝑎𝑛𝑛∞𝑛𝑛=1be a series with nonzero terms. Let 𝜌𝜌= lim𝑛𝑛→∞|𝑎𝑎𝑛𝑛+1𝑎𝑎𝑛𝑛|.i.If 0≤ 𝜌𝜌< 1,then ∑𝑎𝑎𝑛𝑛∞𝑛𝑛=1converges absolutely. ii.If 𝜌𝜌> 1or 𝜌𝜌=∞,then ∑𝑎𝑎𝑛𝑛∞𝑛𝑛=1diverges. iii.If 𝜌𝜌= 1,the test does not provide any information. The Root Test: Let ∑𝑎𝑎𝑛𝑛∞𝑛𝑛=1be a series with nonzero terms. Let 𝜌𝜌= lim𝑛𝑛→∞�|𝑎𝑎𝑛𝑛|𝑛𝑛.i.If 0≤ 𝜌𝜌< 1,then ∑𝑎𝑎𝑛𝑛∞𝑛𝑛=1converges absolutely. ii.If 𝜌𝜌> 1or 𝜌𝜌=∞,then ∑𝑎𝑎𝑛𝑛∞𝑛𝑛=1diverges. iii.If 𝜌𝜌= 1,the test does not provide any information. Examples: 1.Test the convergence of ∑𝑛𝑛3𝑛𝑛∞𝑛𝑛=1. 𝑎𝑎𝑛𝑛=𝑛𝑛3𝑛𝑛,𝑎𝑎𝑛𝑛+1=𝑛𝑛+ 13𝑛𝑛+1Since, lim𝑛𝑛→∞�𝑎𝑎𝑛𝑛+1𝑎𝑎𝑛𝑛�= lim𝑛𝑛→∞𝑛𝑛+13𝑛𝑛+1𝑛𝑛3𝑛𝑛= lim𝑛𝑛→∞𝑛𝑛+13𝑛𝑛+1×3𝑛𝑛𝑛𝑛= lim𝑛𝑛→∞𝑛𝑛+1𝑛𝑛× lim𝑛𝑛→∞3𝑛𝑛3𝑛𝑛+1= 1 ×13=13< 1.Therefore, the given series converges absolutely and thus, converges. 2.Test the convergence of ∑�𝑛𝑛+22𝑛𝑛+3�𝑛𝑛.∞𝑛𝑛=1Let 𝑎𝑎𝑛𝑛=�𝑛𝑛+22𝑛𝑛+3�𝑛𝑛. Since, lim𝑛𝑛→∞�|𝑎𝑎𝑛𝑛|𝑛𝑛= lim𝑛𝑛→∞𝑛𝑛+22𝑛𝑛+3→12< 1. Therefore, the given series converges absolutely and thus, converges.
Math 2212 – Calculus of One Variable II Worksheet 5.6 2Practice Problems1.Use the ratio test to determine whether the given series converges or diverges. State if the ratio test is inconclusive. a.∑1𝑛𝑛! ∞𝑛𝑛=1b.∑𝑛𝑛!10𝑛𝑛∞𝑛𝑛=1c.∑10𝑛𝑛𝑛𝑛!∞𝑛𝑛=1d.∑𝑛𝑛42𝑛𝑛∞𝑛𝑛=1e.∑𝑛𝑛10×10𝑛𝑛𝑛𝑛!∞𝑛𝑛=1f.∑𝑛𝑛!�𝑛𝑛𝑒𝑒�𝑛𝑛∞𝑛𝑛1g.∑(−3)𝑛𝑛(2𝑛𝑛+1)!∞𝑛𝑛=12.Use the Root Test to determine whether the given series converges or diverges. State if the Root Test is inconclusive. a.∑�2𝑛𝑛+13𝑛𝑛−2�𝑛𝑛∞𝑛𝑛=1b.∑�3𝑛𝑛−22𝑛𝑛+1�𝑛𝑛∞𝑛𝑛=1c.∑�2𝑛𝑛2+13𝑛𝑛2−2�𝑛𝑛∞𝑛𝑛=1d.∑�ln 𝑛𝑛𝑛𝑛�𝑛𝑛∞𝑛𝑛=1e.∑�1−2𝑛𝑛4+5𝑛𝑛�𝑛𝑛∞𝑛𝑛=1f.∑(ln 𝑛𝑛)2𝑛𝑛𝑛𝑛𝑛𝑛∞𝑛𝑛=13.Use any test to determine whether the series is absolutely convergent, conditionally convergent, or divergent. a.∑2𝑛𝑛𝑛𝑛×3𝑛𝑛∞𝑛𝑛=1b.∑�1−1𝑛𝑛�𝑛𝑛2∞𝑛𝑛=1c.∑𝑛𝑛×52𝑛𝑛10𝑛𝑛+1∞𝑛𝑛=1d.∑𝑛𝑛43𝑛𝑛∞𝑛𝑛=1e.∑(−1)𝑛𝑛𝑛𝑛∞𝑛𝑛=1f.∑�−𝑛𝑛2𝑛𝑛+1�3𝑛𝑛∞𝑛𝑛=1
Math 2212 – Calculus of One Variable II Worksheet 5.6 3Answers 1.(a) Converges (b) Diverges (c) Converges (d) converges (e) Converges (f) The Ratio Test is inconclusive (g) Converges 2.(a) Converges (b) Diverges (c) Converges (d) converges (e) Converges (f) Converges 3.(a) Absolutely converges (b) Absolutely converges (c) Diverges (d) Absolutely convergent (e) Conditionally convergent (f) Absolutely convergent