MATH 201 Exam 1, Spring 2016

American University of Beirut MATH 201 Calculus and Analytic Geometry III Fall 2008-2009 quiz # 1 1) (15 points) Find the limit of the following sequences: a) 2 ln3 n √n + 1 + 5 b) n sin(1/n) 2 + n2/n c) n√ 3n + 2n 2) (40 points, 8 points each) Determine if the following series converges or diverges. Justify your answers a) +∞ � n=1 �n + 2 n + 1 �n2 b) +∞ � n=0 1 2n(n + 1) c) +∞ � n=0 cos (n2 + 1) n3 + 3n d) +∞ � n=1 (1 − 1 n)n √n + 10 e) +∞ � n=2 1 (ln n)ln n 3) (15 points) Find the sum of the series +∞ � n=2 �3n−2 22n + ln � 1 − 1/n2�� 4) (20 points) Find the interval of convergence of the power series +∞ � n=1 1 √ 2 + n2 .(x + 1)n. For what value(s) of x for which the series converges (i) absolutely? (ii) conditionally? 5) a) (4 points) Using the fact that (tan−1 x)′ = 1 1+x2 , find the Maclaurin series expansion of tan−1 x b) (3 points) express π/4 as a sum of an alternating series. c) (3 points) what is the error done in approximating π/4 with 1 − 1/3 + 1/5 ?