Unformatted text preview: 10/04/2001 THU 12:26 FAX 6434330 MOFFITT LIBRARY I001
George M. Bergman Fall 1999, Math 1B 2 November, 1999
961 Evans Hall Second Midterm — makeup exam 8:10—9:30 AM 1. (36 points, 6 points apiece) Find the following. If an expression is undeﬁned, say so. (a) 2:22 5”/n5. (b) 23:19” + 2%”). (c) The set of all real numbers p such that 22:2 72—13 (In n)_2 converges.
(d) The Maclaurin series for sin 7m. (6) The Taylor series for Ur3 centered at x = — 1. (f) The solution to the differential equation xy’ = y2+1 satisfying the initial condition
31(1) = 1- 2. (16 points) Let a and b be real numbers. Prove that 22:1 converges if ”03+ ”b and only if at least one of a and b is > 1. 3. (30 points, 6 points apiece) For each of the items listed below, give either an example
of the situation described, or a brief reason why no such example exists. (If you give an example, you are not asked to show that it has the asserted property.) (a) A power series 2:0 an (Jr—1)" which converges only at x = 2.
(b) A power series 2:121 an x” which converges for all xe [—1, 1] and no other x. (c) A power series 2:20 an (x+2)n which converges for all real numbers x. (d) A series 2320:] a which converges, but such that 211 [an| diverges. n (e) A series 211 an which diverges, but such that 2;} Ian| converges. 4. (18 points) (a) (7 points) Find the ﬁrst three terms (i.e., the constant, linear, and ——x square terms) of the Taylor series for e centered at x: 2. (b) (11 points) Prove using the formula for the remainder (“Taylor’s Formula”) that for —x all x in the interval [1.5, 2.5], the sum of the above three terms approximates e to within 63/2/48. ...
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