Math 1B Quiz 8GSI: Theo Johnson-Freydhttp://math.berkeley.edu/∼theojf/08Summer1B/Tuesday, 29 July 2008Name: Score: /10You have twenty minutes (plus the break) to complete the following closed-note open-chalkboard quiz. Partial credit will be awarded for correct work, and no points will be givenfor simply writing down the correct answer. Please box your final answers. Use the backof the page as necessary.1. (5 pts) Determine if the following sequence converges or diverges. If it converges,find the limit. If it diverges, explain how you know.(1 +1n2n+s1 + e−ncos n4 −1ln n)∞n=2We assume the sequence converges, and try to compute the limit. If it diverges, wewill presumably fail.limn→∞"1 +1n2n+s1 + e−ncos n4 −1ln n#= limn→∞1 +1n2n+ limn→∞s1 + e−ncos n4 −1ln n=limn→∞1 +1nn2+slimn→∞1 + e−ncos n4 −1ln n= e2+s1 + limn→∞e−ncos n4 − limn→∞1ln n= e2+r1 + 04 − 0= e2+14: the sequence converges2. (0 pts) Can you make the currently schedule office hours times? What’s a time duringthe week for office hours that you would actually attend?Yes, and I’m there every day.3. (5 pts) Determine whether the following series converges or diverges. Explain howyou know.∞Xn=1n! ennn+1Remembering Stirling’s formula, we use the limit comparison test, comparing (n!en)/nn+1with 1/√n:limn→∞(n!en)/nn+11/√n= limn→∞n! ennn√n=√2πwhich is finite and positive. ThusPn! ennn+1converges if and only ifP1/√n does. ButP1/√n diverges by the p-test with p = 1/2, i.e. by the integral test comparing
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