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Exam 2 Practice Problems Part 1: Multiple Choice/ Fill-in Problem 1: Give examples of the following: [A] A sequence that is bounded but not monotonic [B] A sequence that is convergent but not monotonic [C] A divergent series _ an such that limn too an = 0 Problem 2: Which of the following series can we conclude diverge by using the Divergence Test? Write "D" if the series diverges by the Divergence Test and "N" if the test does not apply. M 8 n = 1 In 2 = 0 18 n2 n = 1 2n2 + 4n+ 1 00 2 n = 2 In (n) Problem 3: For each of the following series, indicate an appropriate series to compare it to and then indicate whether the original series will converge or diverge Comparison Series | Converge/ Diverge n + 3 n=1 n2 + 2n+ 1 OO n n=1 Vn3 + 2n 00 2n + n2 n=0 13 + 30 00 nP n=0 Vn2+ 1 Page 1 of 6Math 1132 Exam 2 Problem 4: Consider a general power series centered at a = 2, ) on(x-2)". Suppose that n=0 the series converges when r = 4 and diverges when r = -1. For each of the following series, indicate whether it is convergent "C", divergent "D", or unknown "U" (i.e. not enough information). [i] > Cn n = 0 00 [ii] > (-1)4men n = 0 [iii] n = 0 Problem 5: Express the MacLaurin series for each of the following functions in both sum- mation notation and as an expanded sum: [ A] f(x) = ex [B] f(x) = sin(x) [C] f(x) = cos(x) [D] f(x) = 1 - 2 Problem 6: Determine if the following statements are true (T) or false (F). [A] Whenever a sequence {an} converges to 0, the series _ an converges. [B] Whenever the series _ an converges, the sequence {an} converges to 0. [C] Whenever the series _ an converges, the series _ lanl converges. [D] If an > bn for all n and _ by converges then _ an converges. [E] If {an} is increasing and [an) 0 diverges by the Alternating Series Test. [G] The series _ M diverges by using the comparison test with En V Page 2 of 6Math 1132 Exam 2 Problem 7: When the Ratio Test is applied to the series En- , what is ant!? (A ) (n + 1)2 6 6 6 (B) ( n+ 1 ) 2 (C) - 7 2 6 (D) ( n + 1 ) n (E) ( n + 1 )n 6 Problem 8: What is the smallest N for which the Alternating Series Estimation Theorem tells us that the remainder RN of the Nth partial sum of )(1)' n3/2 satisfies |RN|