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MATH 101 Quiz #5 (v.T2) Last Name: Thursday, March 17 First Name: Grade: Student-No: Section: Very short answer question 1 1 1 1 1 1. 1 mark To what value does the series 1+ + + + + + · · · converge? Simplify 5 25 125 625 3125 your answer completely. Answer: Short answer questions—you must show your work 2. 2 marks Find the solution to the separable initial value problem: dy 4x3 = y , dx e y(0) = log 5. Express your solution explicitly as y = y(x). 3. 2 marks Show that the series ∞ X 2 i=1 answer completely. 2 − 3 i (i + 1)3 converges and find its limit. Simplify your Long answer question—you must show your work 4. 5 marks The nth partial sum of a sequence {an } is known to have the formula sn = 1 + 3n . 3 + 2n (a) Find an expression for an , valid for n ≥ 2. ∞ X (b) Show that the series an converges. (It will help to make the expression from part (a) n=1 a single fraction.) (c) Find the value of the series ∞ X n=1 an .