152fall2007_8MT1 - X n =1 ( n + 1)( n + 2) n ! (d) X n =1 2...

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Last Name Name Student No Department Section Signature : : : : : : : : : : : : : Code Acad.Year Semester Instructors Date Time Duration ATILIM UNIVERSITY Department of Mathematics Calculus II. I. Midterm Math 152 2007-2008 Fall A. ¨ O., ¨ U.A., A.D. 27.10.2007 10.00 90 minutes 6 Questions on 6 Pages Total 100 Points 1 2 3 4 5 6 Question 1 (3 × 6 pts.) (a) Determine whether or not, the sequence { a n } con- verges, where a n = n 2 n - 2 3 n 3 4 n + n n . (b) Using the Sandwich Theorem, evaluate the limit lim n →∞ cos(1 + n ) n .
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(c) Evaluate the limit lim n →∞ (1 + n 2 ) 1 /n . (Hint: Consider ln(1 + n 2 ) 1 /n ) Question 2 (6+8 pts.) Find the sum of the following series. (a) X n =1 ± 1 n - 1 n + 1 ² (b) X n =0 ± 1 2 n + ( - 1) n 5 n ²
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Question 3 (4 × 7 pts.) Test the following series for convergence. (a) X n =1 ln ± n 2 n + 1 ² (b) X n =1 1 n n (1 + n ) (c)
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Unformatted text preview: X n =1 ( n + 1)( n + 2) n ! (d) X n =1 2 n 2 3 n 2 + 2 n + 1 n Question 4 (8 pts.) Using the Integral Test, determine the series X n =1 1 n (1 + ln 2 n ) is convergent or not. Check all conditions of the test. Question 5 (8+ 9pts.) (a) Find the Taylor series expansion of the function f ( x ) = e x-x-1 x 2 about a = 0 by using the Maclaurin series of e x . (b) Find the Taylor polynomial of degree 3 of the function g ( x ) = ln x about a = 1. Question 6 (15 pts.) Find the radius and interval of convergence of the power series X n =1 (4 x-1) n n ....
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152fall2007_8MT1 - X n =1 ( n + 1)( n + 2) n ! (d) X n =1 2...

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