108_11_2009_second_exam

108_11_2009_second_exam - Second Midterm Examination, MAT...

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Unformatted text preview: Second Midterm Examination, MAT 108, Differential Equations. Last Name First Name Student ID Number Remarks: The test contains 5 pages, including the cover sheet. Do not de-staple the test. The use of books, notes, and calculators is not allowed. You can use scratch paper during the exam. However, you cannot submit any work on your scratch paper with the exam. Use both sides of the sheets. Exercise I. Let f be a 2 p- periodic function such that f ( x ) = a + X n =1 a n cos n p x + b n sin n p x , (1) with a , a i and b i , i = 1 , ,n , and p are real numbers. Prove the Parsevals identity 1 2 p Z p- p [ f ( x )] 2 dx = a 2 + 1 2 X n =1 ( a 2 n + b 2 n ) . (2) Show that the Fourier series decomposition of the 2 -periodic function f ( x ) = x/ 2 for- < x < is given by X n =1 (- 1) n +1 n sin( nx ) ,- < x < . (3) Use Parsevals identity to deduce that n =1 1 n 2 = 2 6 , k =1 1 (2 k ) 2 = 2 24 and...
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This note was uploaded on 01/17/2010 for the course MATH 108 taught by Professor Trangenstein during the Fall '07 term at Duke.

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108_11_2009_second_exam - Second Midterm Examination, MAT...

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