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MATH 101 Quiz #5 (v.T1) 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 4 16 64 256 1024 your answer completely. Answer: Short answer questions—you must show your work 2. 2 marks Find the solution to the separable initial value problem: 2x dy = y, dx e y(0) = log 2. Express your solution explicitly as y = y(x). 3. 2 marks Show that the series ∞ X 3 j=1 answer completely. 3 − 2 j (j + 1)2 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 + 2n . 3 + 4n (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 .