Find the Fourier series for the periodic function

[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​

 

[LCKurtz]

< 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.

 

[vela]

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.

 

[LCKurtz]

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.

 

[CricK0es]

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

 

[Ray Vickson]

< 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.

 

[CricK0es]

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.

 

[Ray Vickson]

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]

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