Exam 3, Fall '06

Exam 3, Fall '06 - radius of convergence ( a ) 3 (1 + x ) 2...

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Tufts University Department of Mathematics Exam 3 Math 12 November 13, 2006 12:00-1:20 No books, notes or calculators are allowed. Cross out what you do not want us to grade. You must show all your work in order to received full credit. Please write neatly. You are required to sign you exam book. With your signature, you pledge that you have neither given or received assistance on this exam. Below you may need some of the following series. 1 1 - x = X n =0 x n | x | < 1 e x = X n =0 x n n ! all x sin( x ) = X n =0 ( - 1) n x 2 n +1 (2 n + 1)! all x cos( x ) = X n =0 ( - 1) n x 2 n (2 n )! all x Find the radius and interval of convergence for the following. 1. (12 pts.) X n =1 ( - 1) n ( x + 2) n n 2 n . 2. (12 pts.) X n =1 3 n x n . 3. (12 pts.) X n =1 1 n 2 ( x - 4) n . 4. (16 pts.) Find a power series expansion about a = 0 for each of the following, and be sure to give the
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Unformatted text preview: radius of convergence ( a ) 3 (1 + x ) 2 ( b ) Z 1 1-x 8 dx 5. (14 pts.) Find the Taylor series of f ( x ) = ln( x ) around a = 2. Give your answer in summation notation. 6. (10 pts.) Find the 3rd Taylor polynomial, T 3 ( x ), of f ( x ) = x around a = 9. EXAM CONTINUES ON REVERSE 7. (12 pts.) Use the Alternating Series Estimation Theorem or Taylors Inequality to estimate ln(1 . 1) to within an error of 0 . 001. Write your answer as a single number. ln(1 + x ) = X n =1 (-1) n +1 x n n 8. (12 pts.) Use the table on the front to nd the sum of each of the following series ( a ) X n =0 (-1) n 2 n +1 6 2 n +1 (2 n + 1)! ( b ) X n =0 1 3 n n !...
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Exam 3, Fall '06 - radius of convergence ( a ) 3 (1 + x ) 2...

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