Math 1B: Discussion ExercisesGSI: Theo Johnson-Freydhttp://math.berkeley.edu/~theojf/09Summer1B/Find two or three classmates and a few feet of chalkboard. As a group, try your hand at thefollowing exercises. Be sure to discuss how to solve the exercises — how you get the solution ismuch more important than whether you get the solution. If as a group you agree that you allunderstand a certain type of exercise, move on to later problems. You are not expected to solve allthe exercises: some are very hard.Exercises marked with an § are from Single Variable Calculus: Early Transcendentals for UCBerkeley by James Stewart. Others are my own or are independently marked.Alternating Series and Absolute ConvergenceLet bnbe a positive decreasing sequence: bn≥ bn+1≥ 0 for every n. ThenP(−1)nbnconverges.Moreover, if we truncate the series after then Nth term, and estimateP∞0(−1)nbnby sN=PN0(−1)nbn, then the error of the estimate is at most |bN+1|.A seriesPanconverges absolutely ifP|an| converges. It is a theorem that ifPanis abso-lutely convergent, then it is convergent. But many series are convergent without being absolutelyconvergent. For example, the Alternating Harmonic seriesP∞1(−1)n−1/n converges by the alter-nating series test, butP∞1(−1)n−1/n=P∞11/n is the divergent Harmonic series. A series thatconverges but does not absolutely converge is said to converge conditionally.1. For what values of p does∞Xn=1(−1)n−1np(a) converge absolutely?(b) converge conditionally?(c) diverge?2. For what values of r does∞Xn=1rn(a) converge absolutely?(b) converge conditionally?(c) diverge?3. § Determine whether the following series are absolutely convergent, conditionally convergent,or divergent:(a)∞X1(−1)nnn + 2(b)∞X2(−1)n√n(c)∞X1(−1)n−1ln(n + 4)(d)∞X1(−1)nn√n3+ 2(e)∞X1(−1)n10n(f)∞X1cos πnn2(g)∞X1(−1)n(−1)ncosπn(h)∞X1−n5n(i)∞X1−12nn14. § How many terms of the series would you need to add in order to find the sum to theindicated accuracy?(a)∞Xn=0(−1)n10nn!, |error| < 0.000005 (b)∞Xn=1(−1)n−1ne−n, |error| < 0.05. § Show that the seriesP(−1)n−1bn, where bn= 1/n if n is odd and bn= 1/n2if n is even,is divergent. Why does the alternating series test not apply?6. (a) Find a sequence {an} so thatP∞n=1andiverges, butP∞n=1(an)2converges.(b) Find a sequence {an} so thatP∞n=1anconverges, butP∞n=1(an)2diverges.7. The Riemann ζ function is defined to be the “analytic continuation of” ζ(s) =P∞n=11ns.(a) For what s does the above definition of ζ(s) converge? I.e. what is the domain of theright-hand-side?(b) Prove that when both sides converge, we have:∞Xn=1(−1)n−1ns=1 −12s−1 ∞Xn=11ns!For what s does the left-hand-side converge?(c) Use the above equation to write a formula for ζ(s) that extends the domain to (0, 1) ∪ (1, ∞).(d) When s = 0, explain why the LHS of the above equation is the geometric seriesP∞0rnwith r = −1. Assuming that lims→0P∞1(−1)n−1/ns= lims→0P∞0rn, find ζ(0) = lims→0ζ(s).8. Make sense of the following proof from Proofs without Words: Exercises in Visual Thinkingby Roger B. Nelsen
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