MATH-UA 122 – Spring 2018 – Exam 2 (Version A) You have 110 minutes to complete this exam. There are 10 questions, please make sure that you have exactly 11 pages. Books, notes and electronic devices are not permitted. Show all work, clearly and in order, if you want to get full credit. I reserve the right to take off points if I cannot see how you arrived at your answer (even if your final answer is correct). If a problem is not clear, please ask for clarification. SR 1 (4 points) SR 2 (6 points) SR 3 (3 points) SR 4 (3 points) SR 5 (9 points) SR 6 (16 points) SR 7 (9 points) LR 8 (25 points) LR 9 (10 points) LR 10 (15 points) Total /100 I pledge that I have completed this exam in compliance with the NYU CAS Honor Code. In par- ticular, I have neither given nor received unauthorized assistance during this exam. Name Signature Date 1 Short Response (50 points) 1. Consider the following alternating series: I. ∞∑ n=4 (−1)n √ 2n n √ n− 3 II. ∞∑ n=1 (−1)n 1 (n+ √ n)3/2 III. ∞∑ n=4 (−1)n 1√ n− 3 The following statements are true about which of the above series? (write your answer as I, II, III, or none) (a) The series converges absolutely: (b) The series converges conditionally: (c) The series diverges: 2. Show whether the following sequences are increasing, decreasing, or non-monotonic. Justify your work. (a) an = 1 2n+ 1 n = 1, 2, 3, . . . (b) an = 3n− 1 5n+ 3 n = 1, 2, 3, . . . 2 3. What does the series ∞∑ n=1 ( e pi )n converge to? (a) pi pi − e (b) pi pi + e (c) e pi − e (d) e pi + e (e) The series diverges 4. What does the series 1− ln 2 + (ln 2) 2 2! − (ln 2) 3 3! + . . . converge to? (a) 2 (b) 1/2 (c) sin(ln 2) (d) cos(ln 2) (e) None of the above 5. Do the following sequences converge or diverge? Justify your work. (a) { n2 cos2 n 2n }∞ n=1 (b) { n √ 21+3n }∞ n=1 (c) { n sinn 2n+ 1 }∞ n=1 3 6. Do the following series converge or diverge? Justify your work. (a) ∞∑ n=1 ne−n (b) ∞∑ n=1 n− 2 n √ n (c) ∞∑ n=1 (−1)n−1 e 1/n n (d) ∞∑ n=3 1 n lnn ln(lnn) 4 7. Find the interval and radius of convergence of the following series. (a) ∞∑ n=1 n2xn 2 · 4 · 6 · · · · · (2n) (b) ∞∑ n=2 xn n(lnn)2 5 Longer Response (50 points) 8. Find the power series representation of f(x) about x = 0, and determine the radius of con- vergence. (a) f(x) = 1 x2 + 5x+ 6 6 (b) f(x) = ln [ (1 + x)(1 + x2) ] 7 9. Use series to evaluate the limit. (a) lim x→0 sinx− x 2×3 (b) lim x→0 1− cosx 1 + x− ex 8 10. Consider the function f(x) = x2 sin(x3) (a) Find the Maclaurin series of f(x). (b) Calculate ∫ f(x) dx 9 (c) Approximate ∫ 1 0 f(x) dx with error less than 10−3 10 This page is intentionally left blank for computations 11
