Math 1B Discussion Exercises GSI Theo Johnson Freyd http math berkeley edu theojf 09Summer1B Find two or three classmates and a few feet of chalkboard As a group try your hand at the following exercises Be sure to discuss how to solve the exercises how you get the solution is much more important than whether you get the solution If as a group you agree that you all understand a certain type of exercise move on to later problems You are not expected to solve all the exercises some are very hard Exercises marked with an are from Single Variable Calculus Early Transcendentals for UC Berkeley by James Stewart Others are my own or are independently marked Ratio and Root Tests P The Ratio and Root Tests have a very similar form Let an be a series The Ratio Test begins by asking you to compute limn an 1 an and the Root Test begins by asking you to computer p n limn an It is a theorem that if both these limits exist then they are equal for a given series either one or both may not exist in which case the P corresponding test is inconclusive so let s call the common limit L Then if L 1 the series an converges absolutely and if L 1 then the series diverges If L 1 the tests are inconclusive P P In general when L 1 it is because the series an is comparable to a P P series 1 np although this is certainly not always the case If so then you can decide whetherP an converges P or diverges and hence whether an is absolutely convergent or not Of course if an is absolutely convergent then you can stop testing if it is not absolutely convergent then P it may still be conditionally convergent and a test like the Alternating Series Test may prove that an converges 1 Determine whether the following series are absolutely convergent conditionally convergent or divergent n X X X n n n 1 1 a b 1 c n 4 100 n n3 2 n 1 n 1 n 1 d g X 1 n e1 n n 1 X n 1 n3 2 n nn P e h X sin 4n n 1 X n 2 f 4n 2n n 1 5n i X n2 2n n 1 X n 2 n n ln n n converges for any x What does this say about limn xn n 1 n 3 a It is a fact that lim 1 e Use this fact and the Ratio Test to determine if n n X n the series converges or diverges nn 2 Prove that n 0 xn n n 1 b It turns out that for the above series the limit in the Root Test also exists What is the limit in the Root Test and what does part a say about its value c Find all numbers x such that the Ratio and Root Tests are inconclusive when applied to the series X xn n n 1 1 nn d Use part b to justify the following estimate n nn en e The estimate in part d is the leading order part of Stirling s Formula which says n n n 2 n e or more precisely n n 1 lim n 2 n ne Use this estimate to determine whether the following series converge of diverge i n X e n n 1 nn X nn en n ii iii n 1 X n n n 1 en n For part iii you may assume that nn n en is a monotonic sequence 4 For which positive integers k does the series X n 2 converge kn n 1 P 5 Let an be aPseries with positive terms and let rn an 1 an Suppose that limn rn L 1 so that an converges by the Ratio Test Let Rn be the remainder after n terms i e Rn an 1 an 2 an 2 a If rn is a decreasing sequence and rn 1 1 show by summing a geometric series an 1 that Rn 1 rn 1 an 1 b If rn is an increasing sequence show that Rn 1 L 6 a Let s say that limn an 1 an L 1 Prove that limn an 6 0 and thus prove one part of the Ratio Test Hint Let rn an 1 an then there is some N so that for n N rn L 1 2 K 1 Use this fact to prove that for n N we have an K n aN Since K 1 limn K n aN and so lim an p b Let s say that limn n an L 1 Prove that limn an 6 0 and thus prove one part of the Root Test p c How would you modify these proofs if limn an 1 an limn n an 7 In this exercise you ll prove the other half of the Ratio Test P a Let an be a series with positive terms and let rn an 1 an Suppose that limn rn L 1 and let K L 1 2 Prove that there is some number N such that for n N we have rn K b Conclude that for n N we have an K n aN P c Use the comparison test and a geometric series to prove that n N an converges ConP clude that 1 an converges Hint K 1 you must prove this P d Now let an be a series with possibly negative terms Explain why parts a c prove P that if lim an 1 an 1 then an converges absolutely 8 Modify and repeat the previous exercise to prove the Root Test 2
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