Science:Math Exam Resources/Courses/MATH307/April 2013/Question Section 202 06 (b)
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Question Section 202 06 (b) 

Find using Parseval`s formula. 
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Hint 

Science:Math Exam Resources/Courses/MATH307/April 2013/Question Section 202 06 (b)/Hint 1 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. From part (a) the coefficients of the Fourier series were found to be: This yields: Parseval’s Theorem states: Computing the left side: Computing the right side: Here the sum of odds from to was split into two sums from 1 to and 1 to . It can be observed that the second sum is the same as the first where and since , these two sums are equivalent. To get the expression that we want we can make a substitution of variables to remove the odd restriction in our sum. Let . It can be seen that for k=0,1,2,3,..., n will always be odd. Note: Making this substitution will change summation range. At . So the new summation range will be to . Using the left side that was computed above we get: Rearranging this to get the final answer: 