MTH 201 Final, Fall 2006

Problem 1: (8%) (a) Evaluate the following improper integral � 4 0 x √ 16 − x2 dx (b) Determine the convergence or divergence of � ∞ 2 1 4√ x4 − x dx 1 Problem 2: (12%) Consider the following series ∞ � n=1 ln ( n n + 1) (a) Find S20 and simplify your answer. (b) Find the infinite sum ∞ � n=1 ln ( n n + 1). 2 Problem 3: (20%) Which series converge? Converge conditionally? Diverge? Justify your answer. (a) ∞ � n=1 (sin n) ln n n2 (b) ∞ � n=1 � 1 − 49 n2 �n (c) ∞ � n=1 2nn! nn 3 (d) ∞ � n=1 1 n(1 + ln n)2 Problem 4: (8%) Find the interval of convergence of: ∞ � n=2 (−1)n(x − 2)n 4n ln n 4 Problem 5: (12%) (a) Express the indefinite integral � e−x3 dx as an (alternating) power series. (b) Deduce the definite integral � 0.1 0 e−x3 dx as a series. (c) Find the appropriate value of n for which Sn approximates the definite integral with an error of magnitude less than or equal to 10−11. 5 Problem 6: (8%) Consider f(x) = x2ex. (a) Express f(x) as a power series. (b) Deduce a value for the sum ∞ � n=1 n + 2 n! (You may use term by term integration or differentiation on your result in part (a)). 6 Problem 7: (4%) Find the following limit using Maclaurin series lim n→∞ � n2 ln (1 + 1 n) − n � Problem 8: (8%) Evaluate the following or show that the limit does not exist (a) lim (x,y)→(0,0) x3 + y6 x3 (b) lim (x,y)→(0,0) x2y x2 + y2 7 Problem 9: (6%) Consider the following function f(x, y) = e10y cos (10x) Verify whether f(x, y) satisfies fxx + fyy = 0 (This is called Laplace’s equation) 8 Problem 10: (6%) Consider the following function f(x, y) = � 49 − x2 − y2 (a) Find the domain and range of this function. (b) Find the equation of the level curve of f(x, y) that passes through the point (2, 5). 9 Problem 11: (8%) Evaluate the integral � 1 0 � 1 0 e(2x+3y) dx dy 10