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Math 110 Exam 2 April 10, 2006 Name Show all work of course. 1. Indicate whether the following series are convergent or divergent. Indicate which tests could be used to prove convergence or divergence (you do not need to actually perform the tests). For those series which are convergent find (or estimate) their sum. 1 A) n 1 n! B) n n 0 2 1 1 C) n2 2 n 1 n 1 D) n2 3 n 1 n 1 2. Give examples, if possible. If not possible, say why not. A) A sequence that converges to e B) A series that converges to e C) A sequence {an} such that a n 1 D) A divergent geometric series E) A divergent alternating series n diverges but (1) n 1 n a n converges 3. Use the integral test to prove that the p-series 1 n n 1 p converges if p > 1 but diverges otherwise. Note that p = 1 is a special case and needs to be treated separately. 4. A) Write the Maclaurin series (Taylor series about 0) for f ( x) e x . Show at least four nonzero terms. B) Use the result in part (A) to write the Maclaurin series for f ( x) e x , still showing at least four nonzero terms 2 C) Use the result in part (B) to write an alternating series which converges to 1 e x dx 2 0 1 D) Estimate e x dx by adding or subtracting the first three nonzero terms. 2 0 E) According to the error analysis for alternating series, (not Lagrange’s error bound for Taylor series), what is an upper bound for the error in the approximation of part (D)? 5. A) A function f has f(3) = 1, f ' (3) 5 , and f ' ' (3) 10 . Find the best estimate you can for f(3.1) B) Suppose 10 + 9 + … is a geometric series. What is the sum of the series?