152summer2003_4MT1 - Show that a n is decreasing(b Is a n...

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Last Name Name Student No Department Section Signature : : : : : : : : : : : : : Code Acad. Year Semester Instructors Date Time Duration M E T U Department of Mathematics Calculus-II I.Midterm Math 152 2004 Summer A. ¨ O., M.K., B.K. 16.07.2004 19.00 90 minutes 4 Questions on 4 Pages Total 60 Points 1 2 3 4 Question 1 (6+6+6 pts.) Determine whether the following series are conver- gent or divergent. State the test you used. (a) X n =1 n sin ( 1 n ). (b) X n =1 n 6 / 5 + 1 n 7 / 3 + n 2 + 5 . (c) X n =1 (2 n )! 5 n ( n !) 2
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Question 2 (6+4 pts.) Let a n = (2 n )! 4 n ( n !) 2 .
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Unformatted text preview: Show that { a n } is decreasing. (b) Is { a n } convergent? why? Question 3 (15 pts.) Find the interval of convergence of the power series ∞ X n =1 (-1) n √ n 4 n ( n + 1) ( x-1) 2 n . Question 4 (10+7 pts.) (a) Find the Taylor series of f ( x ) = 1 (5-2 x ) 2 around x = 2. What is the interval of convergence? (b) Find the sum of the series ∞ X n =1 n 2 n-1 ....
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