# Find the Fourier series for the periodic function

• CricK0es
In summary: Use LaTeX to make equations easier to read.Your Fourier Series is correct. So you have$$x^2 = \frac {\pi^2} 3 + \sum_{n=1}^\infty \frac{4(-1)^n}{n^2}\cos(nx)$$on the interval ##[-\pi,\pi]##. Try plugging in the two values to see if you can arrange it to get your results.It's probably just a typo, but the Fourier series should be$$\frac{\pi^2}{3}+\sum_{n=1}^\infty \frac{4(-1)^n}{n^ CricK0es < Mentor Note -- thread moved to HH from the technical forums, so no HH Template is shown > Hi all. I'm completely new to these forums so sorry if I'm doing anything wrong. Anyway, I have this question... Find the Fourier series for the periodic function f(x) = x^2 (-pi < x < pi), f(x) = f(x+2pi) By considering the particular values x = 0 and x = +/- pi prove that​ ∞^Σ (n=1) 1 / n^2 = pi^2 / 6 and ∞^Σ (n=1) (-1)^n / n^2 = -pi^2 / 12 I missed a couple of lectures due to illness so I'm not entirely sure what to do with this. But I have worked through and found the FS itself. Whether it's right I really don't know... = pi^2 / 3 + ∞^Σ (n=1) 4/n (-1)^n Cos(nx) Again, I'm really sorry that it's all a bit messy but I'm still getting use to everything. I would like to learn how to use LaTeX to make equations easier/clearer also. But, regardless, I would really appreciate some guidance on how to proceed through the question. Many thanks​ Last edited by a moderator: CricK0es said: < Mentor Note -- thread moved to HH from the technical forums, so no HH Template is shown > Hi all. I'm completely new to these forums so sorry if I'm doing anything wrong. Anyway, I have this question... Find the Fourier series for the periodic function f(x) = x^2 (-pi < x < pi), f(x) = f(x+2pi) By considering the particular values x = 0 and x = +/- pi prove that​ ∞^Σ (n=1) 1 / n^2 = pi^2 / 6 and ∞^Σ (n=1) (-1)^n / n^2 = -pi^2 / 12 I missed a couple of lectures due to illness so I'm not entirely sure what to do with this. But I have worked through and found the FS itself. Whether it's right I really don't know... = pi^2 / 3 + ∞^Σ (n=1) 4/n (-1)^n Cos(nx) Again, I'm really sorry that it's all a bit messy but I'm still getting use to everything. I would like to learn how to use LaTeX to make equations easier/clearer also. But, regardless, I would really appreciate some guidance on how to proceed through the question. Many thanks​ Your Fourier Series is correct. So you have$$
x^2 = \frac {\pi^2} 3 + \sum_{n=1}^\infty \frac{4(-1)^n}{n^2}\cos(nx)$$on the interval ##[-\pi,\pi]##. Try plugging in the two values to see if you can arrange it to get your results. Last edited: CricK0es It's probably just a typo, but the Fourier series should be$$\frac{\pi^2}{3}+\sum_{n=1}^\infty \frac{4(-1)^n}{n^2}\cos nx.$$The ##n## in the denominator is squared. CricK0es vela said: It's probably just a typo, but the Fourier series should be$$\frac{\pi^2}{3}+\sum_{n=1}^\infty \frac{4(-1)^n}{n^2}\cos nx.$$The ##n## in the denominator is squared. Yes, thanks. Fixed but likely irrelevant given that the OP never returned. epenguin Yeah sorry guys. I thought I'd marked it as solved, and so hadn't been on the forums. I had made a mistake on the denominator but I noticed and fixed it. Thank you both CricK0es said: < Mentor Note -- thread moved to HH from the technical forums, so no HH Template is shown > Hi all. I'm completely new to these forums so sorry if I'm doing anything wrong. Anyway, I have this question... Find the Fourier series for the periodic function f(x) = x^2 (-pi < x < pi), f(x) = f(x+2pi) By considering the particular values x = 0 and x = +/- pi prove that​ ∞^Σ (n=1) 1 / n^2 = pi^2 / 6 and ∞^Σ (n=1) (-1)^n / n^2 = -pi^2 / 12 I missed a couple of lectures due to illness so I'm not entirely sure what to do with this. But I have worked through and found the FS itself. Whether it's right I really don't know... = pi^2 / 3 + ∞^Σ (n=1) 4/n (-1)^n Cos(nx) Again, I'm really sorry that it's all a bit messy but I'm still getting use to everything. I would like to learn how to use LaTeX to make equations easier/clearer also. But, regardless, I would really appreciate some guidance on how to proceed through the question. Many thanks​ If you are going to use plain text, it is better to write your summations as something like sum_{n=1 ..∞} a_n or sum(a_n: n=1..∞), and you can, of course, replace the word "sum" by the symbol Σ. Constructions like your ∞^∑(n=1) a_n are particularly ugly and hard to read, and do not make sense when parsed according to standard rules for reading mathematical expressions. Ray Vickson said: If you are going to use plain text, it is better to write your summations as something like sum_{n=1 ..∞} a_n or sum(a_n: n=1..∞), and you can, of course, replace the word "sum" by the symbol Σ. Constructions like your ∞^∑(n=1) a_n are particularly ugly and hard to read, and do not make sense when parsed according to standard rules for reading mathematical expressions. Thanks, I'll keep that in mind if something else comes up. The next time I post, I'll try and use LaTeX for the equations; although, I need to learn how to use it first. CricK0es said: Thanks, I'll keep that in mind if something else comes up. The next time I post, I'll try and use LaTeX for the equations; although, I need to learn how to use it first. There are tutorials available in this Forum, but perhaps the easiest way is to look at someLaTeX statements of others. For example, you can write ##\pi^3/2 + \sum_{n=1}^{\infty} (-1)^n 4 / n^2 \cos(nx)## or ##\pi^3/2 + \sum_{n=1}^{\infty} (-1)^n 4 / n^2 \cos nx## for an in-line expression and$$\sum_{n=1}^{\infty} (-1)^n \frac{4}{n^2} \cos (nx) \; \text{or} \; \sum_{n=1}^{\infty} (-1)^n \frac{4}{n^2} \cos nx$$for a displayed expression. To see the constructions used, just double right-click on the image and ask for a display of math as tex. The in-line version is initiated and terminated by "# #" (with space between the two #s removed), so # # some formula # #. The displayed version uses " " instead (again, with no spaces between the two s). Note in particular the use of "\cos" instead of "cos"; try it both ways and see why. CricK0es Ray Vickson said: There are tutorials available in this Forum, but perhaps the easiest way is to look at someLaTeX statements of others. For example, you can write ##\pi^3/2 + \sum_{n=1}^{\infty} (-1)^n 4 / n^2 \cos(nx)## or ##\pi^3/2 + \sum_{n=1}^{\infty} (-1)^n 4 / n^2 \cos nx## for an in-line expression and$$\sum_{n=1}^{\infty} (-1)^n \frac{4}{n^2} \cos (nx) \; \text{or} \; \sum_{n=1}^{\infty} (-1)^n \frac{4}{n^2} \cos nx
for a displayed expression.

To see the constructions used, just double right-click on the image and ask for a display of math as tex. The in-line version is initiated and terminated by "# #" (with space between the two #s removed), so # # some formula # #. The displayed version uses "" instead (again, with no spaces between the two \$s). Note in particular the use of "\cos" instead of "cos"; try it both ways and see why.

I see. I'll have a play around with it and see what I can do. Thank you

## 1. What is a Fourier series for a periodic function?

A Fourier series is a mathematical representation of a periodic function as a sum of sinusoidal functions with different frequencies and amplitudes. It allows us to break down a complex periodic function into simpler components.

## 2. How do you find the coefficients for a Fourier series?

The coefficients for a Fourier series can be found by using the Fourier series formula, which involves integrating the periodic function with respect to the sine and cosine functions. The resulting coefficients represent the amplitudes of the individual sinusoidal components in the Fourier series.

## 3. Can any periodic function be represented by a Fourier series?

Yes, any periodic function can be represented by a Fourier series. This is known as the Fourier series convergence theorem, which states that any periodic function with a finite number of discontinuities can be expressed as a Fourier series.

## 4. What is the purpose of finding the Fourier series for a periodic function?

The purpose of finding the Fourier series for a periodic function is to simplify the representation of the function and make it easier to analyze. It also allows us to approximate the function with a finite number of terms, which can be useful in applications such as signal processing and data compression.

## 5. How do you determine the convergence of a Fourier series?

The convergence of a Fourier series can be determined by using the Dirichlet test, which states that if a periodic function has a finite number of discontinuities and is piecewise continuous, then its Fourier series will converge at every point where the function is continuous and will converge to the average of the left and right limits at the points of discontinuity.

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