M257_316_Midterm2_sec103W2006

M257_316_Midterm2_sec103W2006 - Math 257/316, Midterm 2,...

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Math 257/316, Midterm 2, Section 103 17 November 2006 Instructions. The duration of the exam is 55 minutes. Answer all questions. Calculators are not allowed. Maximum score 100. 1. Let f ( x ) be a function that has a period of 2 π having the values: f ( x )= ½ 0 when π x< 0 π x when 0 <x π (a) Determine the Fourier series expansion for f ( x ) . (b) To what value does this series converge when x =0 ? Use this value and the series in (a) to show that: π 2 8 = 1 1 2 + 1 3 2 + 1 5 2 + ... Hint: It may be useful to know that for n 6 : π Z 0 ( π x )cos( nx ) dx = 1 cos( n π ) n 2 π Z 0 ( π x )sin( nx ) dx =
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