MATH 201 Exam 1, Fall 2012

American University of Beirut MATH 201 Calculus and Analytic Geometry III Fall 2012 quiz # 1 Exercise 1 (10 points) Find the limit of the following sequences: a) n2 2n + 1 sin(3/n) b) nn + 1 2n + n! c) ( 1 + 1 3n )2n Exercise 2 (35 points) Determine if the following series converges or diverges Justify your answers a) +∞ ∑ n=1 10n (ln n)n b) +∞ ∑ n=1 1 n2n − 1 c) +∞ ∑ n=1 2 cos(n!) − 1 n(n + 1) d) +∞ ∑ n=2 ln(1 + e3n2) n√n e) +∞ ∑ n=1 (e2/n − 1) Exercise 3 (20 points) a) Find the interval of convergence of the power series +∞ ∑ n=1 (−1)n−1 n4n (3x − 1)2n (do not forget to check at the end points) b) For what value(s) of x the series converges absolutely ? conditionally ? Exercise 4 (15 points) Let f(x) = x − 1 3 + 2x. Find the Taylor series of f about x = 1, then find f(101)(1) Exercise 5 (10 points) Find the following limit: lim x→0 cos(√x) − 1 + x 2 3x2 Exercise 6 (10 points) By using the Maclaurin series of ln(1 + x), give an estimate of ln(1.1) with an error of magnitude less than 10−3