MATH307 April 2013
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Question Section 202 06 (b)
Find using Parseval`s formula.
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From part (a) the coefficients of the Fourier series were found to be:
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: