Math 1B Worksheet 27:Convergence of integrals, series, and sequencesTuesday, 4 November 2007GSI: Theo Johnson-Freydhttp://math.berkeley.edu/∼theojf/Please introduce yourselves to each other, and put your names at the top of a piece ofblackboard. Take turns being the scribe: each of you should have a chance to write on thechalkboard for at least one of the exercises.These exercises are hard — harder than on the homework, quizzes, or exams. Thatmeans that you should spend some time thinking and talking about them; they’re designedto be solved in groups (the best way to learn mathematics). The problems are roughly inorder of increasing difficulty. I don’t expect any group to solve all of them.Don’t forget to draw pictures.1. (a) DoesZ21/2dx(x5− x)1/3converge or diverge?(b) DoesZ∞0dx(x5− x)1/3converge or diverge?2. (a) Show thatZ∞1sin(πx)x2dxconverges.(b) By comparing with the appropriate series and using the alternating series test,show that (a) is in fact still true with the x2replaced by any xpwith p > 0.(c) By comparing with the appropriate series, show thatZ∞1sin(π√x)xdxdiverges.13. (a) If you know thatPa2nconverges, do you know thatPa3nnecessarily converges?Why or why not?(b) In fact, it’s possible forR∞0f(x)2dx to converge but forR∞0f(x)3dx to diverge.Why does this not disprove the similar statement about sums? How are integralsand sums different?(c) On the other hand, if |f(x)| < 1 for every x ≥ 0, use the integral comparisontest and the absolute convergence test to show that ifR∞0f(x)2dx converges,then so doesR∞0f(x)3dx.4. Solve for x:x„x(x(...))«= 2Is your answer reasonable? What numbers could replace “2” in this problem to makethe final answer converge?5. (a) Let’s say that {an} is a positive strictly-decreasing sequences that converges to0, and thatPbnconverges as a series. Show thatXanbnconverges.(b) Conclude that ifPbnconverges, then so doesP(bn)3.(c) On the other hand, find an example of a sequence {bn} wherePbnconvergesbutP(bn)4diverges.6. By analogy with series, define the “Limit Comparison Test” for integrals. I.e. makesense of the idea that if f (x) ≈ g(x), thenR∞0f(x) dx converges if and only ifR∞0g(x) dx converges. In addition to some limit, what else do you have to assumeabout f and g for your test to be true?7. Consider the recurrence relationcn+1=n + 1k(n + 2)cn(a) If cnsatisfies the above relationship for n ≥ 0, what is the radius of convergenceof∞Xn=0cnxn(b) Show that the positive end-point of this interval of convergence diverges, whereasthe lower endpoint converges.(c) Let y(x) =P∞n=0cnxn. Interpret the recurrence relation as a differential equa-tion for y. Solve this differential
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