Math 257/316, Midterm 2, Section 10317 November 2006Instructions.The duration of the exam is 55 minutes. Answer all questions. Calculators are not allowed.Maximum score 100.1. Letf(x)be a function that has a period of2πhaving the values:f(x)=½0when−π≤x<0π−xwhen0<x≤π(a) Determine the Fourier series expansion forf(x).(b) To what value does this series converge whenx=0? Use this value and the series in (a) to showthat:π28=112+132+152+...Hint:It may be useful to know that forn6:πZ0(π−x)cos(nx)dx=1−cos(nπ)n2πZ0(π−x)sin(nx)dx=
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