MATH 201 Exam 2, Fall 2001

American University of Beirut MATH 201 Calculus and Analytic Geometry III Fall 2005-2006 quiz # 1 Name: ............................... ID #: ................................. 1. (12 points, 6 points each) Find each of the following limits: a) lim n→+∞ n2/n sin2 n √n + 10 b) lim n→+∞ (n + 1)n+1 (2n + 1)nn 2. (40 points, 8 points each) Determine if the following series converges or diverges. Justify your answers a) +∞ � n=0 1 e2n + n b) +∞ � n=0 n2ne−n c) +∞ � n=1 ln3 n n√n d) +∞ � n=1 (−1)n cos n 2n e) +∞ � n=1 1 nln n/n 3. (14 points) Find ∞ � k=1 � k! − 6k 3k.(k − 1)! � 4. a) (15 points) Find the interval of convergence of the power series +∞ � n=2 (2x − 1)n 4n ln n (be sure to check convergence at the endpoints) b) (3 points) For what value(s) of x for which the series converges (i) absolutely? (ii) conditionally? 5. Let f(x) = xe−x2. a) (5 points) Find the Maclaurin series expansion of f. b) (5 points) How accurate is the approximation f(x) = x − x3 on the interval [0; 0.1] ? c) (6 points) Find f(n)(0). (hint: you may notice that f is odd !)