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 Alternating Series and Absolute Convergence P Let bn be a positive decreasing sequence bn bn 1 0 for every n ThenP 1 n bn converges n Moreover if we truncate the series after then N th term and estimate 0 1 bn by sN PN n then the error of the estimate 0 1 bn P Pis at most bN 1 P A series an converges absolutely if an converges It is a theorem that if an is absolutely convergent then it is convergent But many series are convergent without being absolutely P n 1 convergent For example Harmonic series 1 1 n converges by the alterP the Alternating P n 1 n 1 n is the divergent Harmonic series A series that nating series test but 1 1 1 converges but does not absolutely converge is said to converge conditionally 1 For what values of p does X 1 n 1 np n 1 a converge absolutely b converge conditionally c diverge 2 For what values of r does X rn n 1 a converge absolutely b converge conditionally c diverge 3 Determine whether the following series are absolutely convergent conditionally convergent or divergent a d X 1 n n 1 X 1 g X 1 b n 2 1 n n n3 2 1 n 1 n cos n e X 1 n n 2 X 1 1 n 10n X n n h 5 1 1 c X 1 n 1 ln n 4 1 X cos n n2 1 X 1 n i 2n f 1 4 How many terms of the series would you need to add in order to find the sum to the indicated accuracy a X 1 n n 0 10n n error 0 000005 X b 1 n 1 ne n error 0 0 n 1 P 5 Show that the series 1 n 1 bn where bn 1 n if n is odd and bn 1 n2 if n is even is divergent Why does the alternating series test not apply P P 2 6 a Find a sequence an so that n 1 an diverges but n 1 an converges P P 2 b Find a sequence an so that n 1 an diverges n 1 an converges but P 1 7 The Riemann function is defined to be the analytic continuation of s n 1 ns a For what s does the above definition of s converge I e what is the domain of the right hand side b Prove that when both sides converge we have X 1 n 1 n 1 ns 1 X 1 2s 1 ns 1 n 1 For what s does the left hand side converge c Use the above equation to write a formula for s that extends the domain to 0 1 1 P n d When s 0 explain why the LHS of the above equation is the geometric series 0 r P n P n 1 s n lim 0 r find 0 lim s with r 1 Assuming that lim 1 1 s 0 s 0 s 0 8 Make sense of the following proof from Proofs without Words Exercises in Visual Thinking by Roger B Nelsen 1993 2
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