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exam_3_s07 - is convergent find its limit 2 Let p n be the...

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Math 1312 Spring 2007 Calculus II Third Midterm Exam Tuesday, April 3, 7:00 PM - 9:00 PM Your name (please print): Instructions: Other than a single 8.5” × 11” page of handwritten notes, this is a closed book, closed notes exam. Use of calculators is not permitted. You must justify all of your answers to receive credit. Notation is important, and points will be deducted for incorrect use. Please do all of your work on the paper provided, work each problem on a separate sheet of paper, and staple your pages in the correct order when you are finished. The Honor Code requires that you neither give nor receive any aid on this exam. If you are bound by the Academic Honor Code, please indicate that you have read and understood these guidelines by signing your name in the space provided: Pledged: Do not write below this line Problem 1 2 3 4 5 Points 10 10 10 10 60 Score Total:
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1. Define the sequence { a n } by setting a 1 = 1 and a n +1 = 1 1 + a n for n 1. Assuming that
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Unformatted text preview: is convergent, find its limit. 2. Let { p n } be the sequence of prime numbers, i.e. { p n } = { 2 , 3 , 5 , 7 , 11 , 13 , 17 , 19 , 23 , 29 , . . . } . Show that ∞ X n =1 (-1) n +1 p n converges. 3. Find a power series representation for the function g ( x ) = x x-5 in powers of x and state for which values of x this representation is valid. 4. Find the interval of convergence of the power series ∞ X n =1 x n 3 n n 1 / 2 . Don’t forget to check for convergence at the endpoints of the interval! 5. Determine if the series converges or diverges. If it converges, determine if this convergence is absolute or conditional. a. ∞ X n =1 (-1) n n + 1 n 3-3 n 2 + 9 n + 13 b. ∞ X n =1 ± n 2 + 1 2 n 2 + 1 ² n c. ∞ X n =2 1 n (ln n ) 2 d. ∞ X n =1 (-1) n +1 4 √ n + 1 e. ∞ X n =1 (-1) n n n + 5 f. ∞ X n =1 1 5 + (-3) n Calculus II, Exam 3 Work Page...
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