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Find at least three nonzero terms (including $a_{0}$ and at least two cosine terms and two sine terms if they are not all zero) of the Fourier series for the given functions, and sketch at least three periods of the function.$$f(x)=x \quad-\pi \leq x<\pi$$

Calculus 2 / BC

Chapter 30

Expansion of Functions in Series

Section 6

Introduction to Fourier Series

Series

Campbell University

Baylor University

University of Nottingham

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to find the four year Siris of f of X equals three x will start by evaluating what a subzero is and this is going to be equal to We'll call these limits negative pie el toe positive pie el And so it would be, um, won over l which in this case is pie integrated from negative pie or negative L. A positive L and we'll evaluate this limit of this integral from three x dx or of the re X dx. And then from here This would actually, since it's odd, the area to the left will cancel with the area to the right, so this would actually evaluate to zero. So this first term will be zero from here. Well, you could go ahead and switch over to a sub k. Oh, go and do that underneath and bread and it's up, K. And it's going to be cool, the integral one over El or one of her pie from negative pi to pie again. And then it's just going to multiply. Our function reacts Times Co sign So three x co sign K X and using Wolf Rammer. Dez most we could pull again this integral for K is one and we get zero and K is to and so on and on for each k value that we pull again, they would all evaluate 20 Solve these terms here will go to zero, Which makes sense because co sign, uh would help model things that are even functions. But our function three x is odd. So we really only want to rely on the last terms which are odd. Help us with this so we'll go ahead and now evaluate. Be sub K. What about switched? I will go black here, so be sub Okay, It's very similar to a so okay except ah replace the co sign with a sign So one over pi from negative pie The pie three x towns sign of X but that k constant in there as well. The axe has left off with the exits for both of those, but they should be there too. And then we'll go ahead and evaluate these. So now that we know that this is odd and it should result in a non zero values if we just replace K with one and then Eudes full frame or Desmond's to evaluate it will end up evaluating that, um, integral in the area under the curve. And it would end up being six when K is one when K is, too, or, in other words, that be sub to, um, value would be negative. Three. That being said three, you could see that these are actually going to alternate between negatives and positives. The sub three is to Sorry, I was looking ahead. So be Step three is to and then finally be sub four. What the negative 1.5. And we could rewrite the, um, term three. X as a four year Siri's with something of signs and CO signs as f of X equals will go and plug in R B sub one value right, that would go in here and it would be six. And then sign K is one. So this would be a woman. So six sign one x or just six sine x. And we ignored the other two terms because they evaluated at zero and our next term be sub to evaluated. Here. It's buying this three, and now we're just gonna increase the K value right, because it's now too. And so would be both blood by sign of two X. So hopefully you're starting to see this pattern. Yeah. And each term is just going to increase the, uh, values. Buy one for K each time here. And so next we would have waas. He's up three, which is to be plus two sign of forex. Excuse me a sign of three X and eventually we're gonna draftees from negative to pi to pie so you could go ahead and do that. Now, if you would like toa kind of verify what this would look like or how close toe the function. Three X That's going to be what's just four terms and then the last one minus one point by sign up for X. And so this is our four year Siri's out to four terms. We'll go ahead and expanded up to eight terms in just a moment, but for now, but we'll go ahead. Graft. That's using, uh, graphing tool of your choice. And as we do so, it would look something along the lines of this from negative two pi to pie. It would come up. Down? Uh, no. Okay, so something like this and it's going to be a decent model for this line and blue, which is why equals three X And we know that this Siri's approximation is gonna be better with more turns. So you could test these and draftees on your own, and I'll show you what it looks like for the next four terms. Out to eat, be some five is gonna about in a girl with changing Kato five is gonna be 1.2. Changing Keita six or be sub sects is going to be thank you for he's up. Seven. It's 67 and finally beats up eight but the negative three sports And so these air just the coefficient, basically. And so from here, let's say plus, then we could use this one. It would be plus 1.2 sign of five x minus. Um, you know, minus one sign of six X plus 6/7. Sign of Southern X, minus 3/4 sign of eight X And then if we graft that we would just give us. It's like better approximation for that function. Three X and so it looks very similar to what we just craft, except it's a little a little smoother, even though it. Also, it's a little more. It kind of hugs the line a little better. So again, that's just gonna, um, have reduced a little bit of the error vertically that the previous one did not take care of. And so this is out to eight terms here. And, you know, if we went further and further are approximations would just keep getting better and better. Yeah.

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