1014mockC - n →∞ S n ≤ d 4[6 Does the following...

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[COMMENT: This test is a bit longer than an actual test will be, but the questions and format are typical of what you can expect.] MATH 1014 3.00 MW – APPLIED CALCULUS II – Prof. Madras Practice Test #3 INSTRUCTIONS: Answer all six questions. Justify your answers; i.e. show your work. No calculators or other aids are permitted. Total score: 50 points. Time: 50 minutes. 1. [10] Do the following series converge or diverge (explain): ( a )[5] X n =1 n 2 2 n ( b ) [5] X n =1 ( - 1) n e n - 1 e n + 1 2. [6] Show that the following series converges: X j =1 1 j 3 + j 3. [9] For n = 1 , 2 , . . . , let S n be the partial sum n X k =1 ( - 1) k +1 k . (a) [5] Explain why lim n →∞ S n exists. (b) [4] Using the fact that S 80 = 0 . 54917, find a lower bound and an upper bound for lim n →∞ S n . (That is, find numbers c and d such that c
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Unformatted text preview: n →∞ S n ≤ d .) 4. [6] Does the following series converge or diverge (explain): 1 1 + 1 1 2 + 1 2 + 1 2 2 + 1 3 + 1 3 2 + ··· + 1 n + 1 n 2 + ··· 5. [11] (a) Find the interval of convergence of the power series ∞ X n =1 ( x-3) n ( n + 1)7 n . (b) Write a power series which is the derivative of the power series given in part (a). 6. [8] We know that e z = ∑ ∞ n =0 z n n ! for every real number z . (a) Use the Maclaurin series for e x to find a power series for the function f ( x ) = x 2 e x 2 . (b) Use the result of part (a) to find f (8) (0), the eighth derivative of x 2 e x 2 at x = 0. (Hint : You don’t need to diFerentiate to answer this question.)...
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