Calculus II Homework: The Comparison Tests Page 1. a n. 1 n 2 + n + 1. n= n. n=1

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1 Calculus II Homework: The Compariso Tests Page Questios l coverget or diverget? coverget or diverget? Example Suppose a ad b are series with positive terms ad b is kow to be coverget. a If a > b for all, what ca you say about a? Why? b If a < b for all, what ca you say about a? Why? Example Determie whether the series coverges or diverges coverget or diverget? Example Determie whether the series coverges or diverges. +. Example Suppose that a ad b are series with positive terms ad b is coverget. Prove that if a lim = 0 b the a is coverget. Note: This is a more difficult problem tha most, sice it is a proof that ivolves the defiitio of limit. Example Suppose that a ad b are series with positive terms ad b is diverget. Prove that if a lim = b the a is diverget. Note: This is a more complicated problem tha most, ad ivolves usig a proof by cotradictio. Solutios l coverget or diverget? Let s try to use the Compariso Test. How do we kow what series to compare to? Well, we try somethig, ad use a series which we kow somethig about. We usually try to pick our compariso series based o attributes of the give

2 Calculus II Homework: The Compariso Tests Page series. Sice l > for 3 a = l > = b for 3 =3 is diverget it is a p-series with p =, we kow coverget or diverget? Let s try to use the Limit Compariso Test. For large, l is diverget by the compariso test So let s take a = b = 3 = 3 The series b = 3 is a coverget geometric series sice a = /3, r = /3 < a lim b / + 5/ / = > 0 ad fiite. Sice b coverges, a coverges by the limit compariso test. Example Suppose a ad b are series with positive terms ad b is kow to be coverget. a If a > b for all, what ca you say about a? Why? b If a < b for all, what ca you say about a? Why?

3 Calculus II Homework: The Compariso Tests Page 3 a If a > b for all, ad b is coverget, the we caot say aythig about a sice it is ot bouded above by b. b Sice a is positive, the series a must be icreasig sice we are always addig a positive quatity to the partial sum i other words, s + > s. If a < b for all, ad b is coverget, the a is coverget sice it is bouded above by b which coverges. Example.4.3 Determie whether the series coverges or diverges Let s try to use the Compariso Test. Let s try to pick our compariso series based o attributes of the give series. + + > for a = < + + = b for ote chage i the relatio Sice b = compariso test. is coverget it is a p-series with p =, we kow a = + coverget or diverget? Let s try to use the Limit Compariso Test. For large, + + is coverget by the + So let s take a = + b = = The series b = a lim b is a coverget geometric series sice a = /, r = / < / = > 0 ad fiite.

4 Calculus II Homework: The Compariso Tests Page 4 Sice b coverges, a coverges by the limit compariso test. Example Determie whether the series coverges or diverges. +. Let s try to use the Compariso Test. Let s try to pick our compariso series based o attributes of the give series. + > for a = + < = b for ote chage i the relatio Sice b = is diverget it is a p-series with p = /, this does t tell us aythig about a see... Sice this does t help us, we ll have to try somethig else. Let s try the limit compariso test with the compariso series b = a + lim b / = > 0 ad fiite.. Sice b diverges, a diverges by the limit compariso test. Example Suppose that a ad b are series with positive terms ad b is coverget. Prove that if a lim = 0 b the a is coverget. Note: This is a more difficult problem tha most, sice it is a proof that ivolves the defiitio of limit. a Sice lim = 0, by the defiitio of limit we kow there exists a N > 0 such that a /b 0 < for all > N. b Sice a ad b are positive, a /b 0 < a < b. Therefore, sice b coverges, a coverges by the compariso test.

5 Calculus II Homework: The Compariso Tests Page 5 Example Suppose that a ad b are series with positive terms ad b is diverget. Prove that if a lim = b the a is diverget. Note: This is a more complicated problem tha most, ad ivolves usig a proof by cotradictio. Assume a coverges. a b Sice lim = lim = 0. b a Usig the result from Problem.4.40 a, we kow that if a coverges the b coverges as well. But we are told that b diverges cotradictio. diverges. Therefore, the assumptio we made must be wrog, ad a