MATH 201 Exam 1, Fall 2006

American University of Beirut MATH 201 Calculus and Analytic Geometry III Fall 2006-2007 quiz # 1 Name: ............................... ID #: ................................. 1. (15 points, 5 points each) Find each of the following limits: a) lim n→+∞(−1)n n n + 1 + 1 b) lim n→+∞ ln10 n n + 10 n2/n c) lim n→+∞ �3n + 1 3n �2n−1 2. (10 points) Show that the sequence an = (n!)2 4n (2n)! is nondecreasing. What can you say about the series +∞ � n=0 (n!)2 4n (2n)! ? justify. 3. (30 points, 10 points each) Determine if the following series converges or diverges. Justify your answers a) +∞ � n=0 ln n + en 22n + n3 b) +∞ � n=1   e � 1 + 1 2n �n   n c) +∞ � n=2 1 nln n 4. (15 points) Find ∞ � n=0 � (−1)n π2n+1 4n(2n + 1)! + 23n 32n+2 � 5. a) (18 points) Find the interval of convergence of the power series +∞ � n=1 (x + 1)n 1 + ln n (be sure to check convergence at the endpoints) b) (2 points) For what value(s) of x for which the series converges (i) absolutely? (ii) conditionally? 6. (10 points) Show that for x > 0, 1 − x 2 < ln(1 + x) x < 1. (hint: use the Maclaurin series of ln(1 + x) and the Alternating Estimation Theorem)