MATH307 December 2010
• Q1 (a) • Q1 (b) • Q1 (c) • Q1 (d) • Q1 (e) • Q2 (a) • Q2 (b) • Q2 (c) • Q2 (d) • Q2 (e) • Q3 (a) • Q3 (b) • Q3 (c) • Q3 (d) • Q4 (a) • Q4 (b) • Q5 (a) • Q5 (b) • Q5 (c) • Q6 (a) • Q6 (b) • Q6 (c) • Q6 (d) • Q6 (e) • Q6 (f) • Q7 (a) • Q7 (b) • Q7 (c) •
Question 06 (c)
Explain how the orthogonality of the functions allows you to relate the inner product in part (b) to the sum
Use your answer to to calculate the inﬁnite sum
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
Use Parseval's identity
and your results from part (a) and part (b).
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According to Parseval's identity it holds that
where cn are the Fourier coefficients of ƒ(x). In part (b) we calculated the right hand side and found that the above equals to 1/3.
In part (a) we found that the Fourier coefficients are and . Hence
where we have noted that in splitting up the sum. Solving for the remaining series yields the final answer