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

From UBC Wiki

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: