Math 1B Worksheet 27 Convergence of integrals series and sequences Tuesday 4 November 2007 GSI Theo Johnson Freyd http math berkeley edu theojf Please introduce yourselves to each other and put your names at the top of a piece of blackboard Take turns being the scribe each of you should have a chance to write on the chalkboard for at least one of the exercises These exercises are hard harder than on the homework quizzes or exams That means that you should spend some time thinking and talking about them they re designed to be solved in groups the best way to learn mathematics The problems are roughly in order of increasing difficulty I don t expect any group to solve all of them Don t forget to draw pictures 1 a Does 2 Z dx 1 2 x5 x 1 3 converge or diverge b Does Z dx x5 0 x 1 3 converge or diverge 2 a Show that Z 1 sin x dx x2 converges b By comparing with the appropriate series and using the alternating series test show that a is in fact still true with the x2 replaced by any xp with p 0 c By comparing with the appropriate series show that Z sin x dx x 1 diverges 1 P P 3 a If you know that a2n converges do you know that a3n necessarily converges Why or why not R R b In fact it s possible for 0 f x 2 dx to converge but for 0 f x 3 dx to diverge Why does this not disprove the similar statement about sums How are integrals and sums different c On the other hand if f x 1 for every x 0 use the comparison R integral 2 test and the Rabsolute convergence test to show that if 0 f x dx converges then so does 0 f x 3 dx 4 Solve for x x x x 2 Is your answer reasonable What numbers could replace 2 in this problem to make the final answer converge 5 a Let s say that P an is a positive strictly decreasing sequences that converges to 0 and that bn converges as a series Show that X an bn converges P P b Conclude that if bn converges then so does bn 3 P c On the hand find an example of a sequence bn where bn converges P other but bn 4 diverges 6 By analogy with series define the Limit Comparison Test for integrals I e make R f x dx converges if and only if sense of the idea that if f x g x then 0 R g x dx converges In addition to some limit what else do you have to assume 0 about f and g for your test to be true 7 Consider the recurrence relation cn 1 n 1 cn k n 2 a If cn satisfies the above relationship for n 0 what is the radius of convergence of X cn xn n 0 b Show that the positive end point of this interval of convergence diverges whereas the lower endpoint converges P n c Let y x n 0 cn x Interpret the recurrence relation as a differential equation for y Solve this differential equation 2
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