- #1
Winzer
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Homework Statement
Prove:
[tex] f(t)=\sum_{i=1}^\infty (-1)^{n+1} \frac{Sin(n\omega t)}{n} [/tex]
represents the sawtooth function.
Homework Equations
Fourier Series Equations:
[tex] f(t)=\frac{a_0}{2}+\sum_{i=0}^\infty \left(a_n Cos(nt)+b_n Sin(nt) \right) [/tex]
where
[tex] a_n=\frac{2}{T}\int_{t}^{t+T} dt cos(nt) f(t) [/tex]
[tex] b_n=\frac{2}{T}\int_{t}^{t+T} dt sin(nt) f(t) [/tex]
[tex] a_0=\frac{2}{T}\int_{t}^{t+T} dt f(t) [/tex]
The Attempt at a Solution
I see from wikipedia:http://en.wikipedia.org/wiki/Fourier_series that they get
[tex] f(t)=2 \sum_{i=1}^\infty (-1)^{n+1} \frac{Sin(n\omega t)}{n} [/tex]
Is this 2 out in front because of their series is twice the period (T) of the one I'm given?
Anyways. The way I go about this is to first take out the [tex]a_n[/tex] since this is an odd function. The part I am stuck on is the limits of integration. The period is the time it takes to travel from the bottom of the sawtooth wave to the top of one right? So should T be (in the case of my given equation) [tex]-\frac{\pi}{2}[/tex] to [tex]\frac{\pi}{2}[/tex]?