Parseval Relation/Fourier Series

[noranne]

Homework Statement

Slope-matched parabolic sections. Consider the function of period 4 defined over the interval [-2,2] by the equations:

f(t) = 2*t-t^2 for 0<t<2 and f(t) = 2*t+t^2 for -2<t<0

It has a Fourier expansion $$\sum_{m=0}^\infty \frac{32}{\pi^3*(2m+1)^3} sin((2m+1) \frac{\pi}{2} t)$$

Use Parseval's relation to compute the sum $$\sum_{m=0}^\infty \frac{1}{(2m+1)^6}$$

Use the extended Parseval's relation and Fourier series calculated in this handout to compute the sums $$\sum_{m=0}^\infty \frac{1}{(2m+1)^4}$$ and $$\sum_{m=0}^\infty \frac{(-1)^m}{(2m+1)^5}$$

Answers: $$\frac{\pi^6}{960} ; \frac{\pi^4}{96} ; \frac{5 \pi^5}{1536}$$

Homework Equations

Parseval's Relation/Parseval's Extended Relation

The Attempt at a Solution

Okay, we got the first part, the pi to the six over 960. What we don't get is how to use the extended Parseval relation to find the second parts. We tried writing up a Fourier series for an antisymmetric square wave, but we weren't getting anywhere with that. Basically, we're clueless on the second two.

Any ideas?

 

[EnumaElish]

Writing out how you solved the first part as well as the P.E.R. might help to elicit responses.

 

[noranne]

We got it...turned out we were just confused by the phrasing, and the problem was actually really easy. Go figure. Thanks!

 

[EnumaElish]

Always glad to help :)