Convergence to pi^2/6 using Fourier Series and f(x) = x^2

[libelec]

Homework Statement

Using the Fourier trigonometrical series for $$f(x) = {x^2},{\rm{ }}0 \le x < 2\pi$$, prove that $$\sum\limits_{n = 1}^\infty {\frac{1}{{{n^2}}}} = \frac{{{\pi ^2}}}{6}$$

The Attempt at a Solution

This is more of a "what am I doing wrong question". First, because I'm not in the period defined by the trigonometrical coefficients, I have to change the limits using that:

$$0 \le x < 2\pi$$
$$- \pi \le x - \pi < \pi$$
$$\begin{array}{l} - \pi \le t < \pi , \\ t = x - \pi , \\ f(t) = f(x - \pi ) \\ \end{array}$$

Then I find the Fourier coefficients with this in hand:

$${a_0} = \frac{1}{\pi }\int\limits_{ - \pi }^\pi {f(t).dt = } \frac{1}{\pi }\int\limits_0^{2\pi } {f(x).dx = } \frac{1}{\pi }\int\limits_0^{2\pi } {{x^2}.dx = } \frac{8}{3}{\pi ^2}$$

$${a_n} = \frac{1}{\pi }\int\limits_{ - \pi }^\pi {f(t).\cos (nt)dt = } \frac{1}{\pi }\int\limits_0^{2\pi } {f(x).\cos (n(x - \pi ))dx = } \frac{1}{\pi }\int\limits_0^{2\pi } {{x^2}.\left[ {\cos (nx)\cos (n\pi ) - \sin (nx)\sin (n\pi )} \right].dx = }$$
$${a_n} = \frac{1}{\pi }\int\limits_0^{2\pi } {{x^2}.\cos (nx).{{( - 1)}^n}dx = } \frac{{{{( - 1)}^n}}}{\pi }\left( {\left. {\frac{{2x.\cos (nx)}}{{{n^2}}}} \right|_0^{2\pi }} \right) = \frac{{2{{( - 1)}^n}}}{{{n^2}}}$$

$${b_n} = \frac{1}{\pi }\int\limits_{ - \pi }^\pi {f(t).\sin (nt).dx = } \frac{1}{\pi }\int\limits_0^{2\pi } {f(x).\sin (n(x - \pi )).dx = } \frac{1}{\pi }\int\limits_0^{2\pi } {{x^2}.\left[ {\sin (nx)\cos (n\pi ) + \sin (n\pi )\cos (nx)} \right].dx = }$$
$${b_n} = \frac{{{{\left( { - 1} \right)}^n}}}{\pi }\int\limits_0^{2\pi } {{x^2}.\sin (nx).dx = } \frac{{{{\left( { - 1} \right)}^n}}}{\pi }\left( {\left. {(\frac{2}{{{n^3}}} - \frac{{{x^2}}}{n})\cos (nx)} \right|_0^{2\pi }} \right) = \frac{{4{{\left( { - 1} \right)}^{n + 1}}}}{n}$$

Then, using Parseval's identity:

$${\left\| {f(x)} \right\|^2} = \frac{{{a_0}^2}}{2} + \sum\limits_{n = 1}^\infty {{a_n}^2 + {b_n}^2}$$
$$\frac{1}{\pi }\int\limits_0^{2\pi } {{{({x^2})}^2}.dx} = \frac{{{{\left( {\frac{8}{3}{\pi ^2}} \right)}^2}}}{2} + \sum\limits_{n = 1}^\infty {{{\left( {\frac{{2{{( - 1)}^n}}}{{{n^2}}}} \right)}^2} + {{\left( {\frac{{4{{( - 1)}^{n + 1}}}}{n}} \right)}^2}}$$
$$\frac{{32}}{5}{\pi ^4} = \frac{{32{\pi ^4}}}{9} + \sum\limits_{n = 1}^\infty {\frac{4}{{{n^4}}} + \frac{{16}}{{{n^2}}}}$$
$$\frac{{112}}{9}{\pi ^4} = \sum\limits_{n = 1}^\infty {\frac{{4 + 16{n^2}}}{{{n^4}}}}$$

And here I reach a deadpoint. What do I do?

 

[LCKurtz]

You are extending f(x) periodically with period $2\pi$. You get the same thing when you integrate a periodic function over any period. That is, if g(x) has period P, then

$$\int_a^{a+P} g(x)\, dx = \int_b^{b+P} g(x)\, dx$$

That means you don't have to use $-\pi$ to $\pi$ in your formulas for the coefficients; you can use 0 to $2\pi$, so you don't have to fiddle with changing the x².

The other thing you are missing is using the Dirichlet theorem which tells you what the series will converge to at 0 and $2\pi$. Notice that your periodic extension of your function has a jump discontinuity at those points.

 

[LCKurtz]

Also, one other question. Are you sure your function x² wasn't given on $(-\pi,\pi)$ in the first place?

 

[libelec]

You are extending f(x) periodically with period $2\pi$. You get the same thing when you integrate a periodic function over any period. That is, if g(x) has period P, then

$$\int_a^{a+P} g(x)\, dx = \int_b^{b+P} g(x)\, dx$$

That means you don't have to use $-\pi$ to $\pi$ in your formulas for the coefficients; you can use 0 to $2\pi$, so you don't have to fiddle with changing the x².

The other thing you are missing is using the Dirichlet theorem which tells you what the series will converge to at 0 and $2\pi$. Notice that your periodic extension of your function has a jump discontinuity at those points.

Let me see if I get it: you're saying that x^2.cos(nx) and x^2.sin(nx) are 2Pi periodic, so that change of limits that produced the (-1)^n was wrong? I didn't have to do anything, just integrate them between 0 and 2Pi?

And what do you want me to do with the borders? I know that they will converge to the lateral limits average but I don't know what you mean.

And yes, x^2 was defined in (0, 2Pi).

 

[LCKurtz]

Let me see if I get it: you're saying that x^2.cos(nx) and x^2.sin(nx) are 2Pi periodic, so that change of limits that produced the (-1)^n was wrong? I didn't have to do anything, just integrate them between 0 and 2Pi?

No, I am not saying that x²cos(nx) and x²sin(nx) are periodic. I am saying that the periodic extensions of them are periodic, and that is what you are expanding in a FS. I didn't go through your steps to see if you had any mistakes an your calculation of a_n and b_n. I am just saying you can use the usual formulas on any period you wish, and in the present case, it is much easier to use $(0,2\pi)$.

And what do you want me to do with the borders? I know that they will converge to the lateral limits average but I don't know what you mean.

And yes, x^2 was defined in (0, 2Pi).

The reason I asked this is because, if I recall correctly, evaluating the even extension of x² on $(-\pi,\pi)$ at 0 is the way I have seen the problem done.

 

[libelec]

No, I am not saying that x²cos(nx) and x²sin(nx) are periodic. I am saying that the periodic extensions of them are periodic, and that is what you are expanding in a FS. I didn't go through your steps to see if you had any mistakes an your calculation of a_n and b_n. I am just saying you can use the usual formulas on any period you wish, and in the present case, it is much easier to use $(0,2\pi)$.

But still, the expressions for a_n and b_n, etc, are made for (-$$\pi$$, $$\pi$$), I still have to make some correction when I calculate them for (0, 2$$\pi$$), right?

 

[LCKurtz]

No. You don't have to change anything but the limits. If f(x) has period $2\pi$ then

$$\frac 1 {\pi}\int_{-\pi}^{\pi} f(x)\cos{nx}\, dx = \frac 1 {\pi}\int_{0}^{2\pi} f(x)\cos{nx}\, dx$$

and ditto for the sine. In the current problem the latter is much preferred because it is the only one in which you can use $f(x) = x^2$.

And looking ahead, I'm guessing you might want to use the even periodic extension of x², which would be $4\pi$ periodic. As I mentioned before, the usual series for this is the periodic extension of x² on $(-\pi,\pi)$, but that is apparently not what you have been asked to do.

 

[libelec]

No. You don't have to change anything but the limits. If f(x) has period $2\pi$ then

$$\frac 1 {\pi}\int_{-\pi}^{\pi} f(x)\cos{nx}\, dx = \frac 1 {\pi}\int_{0}^{2\pi} f(x)\cos{nx}\, dx$$

and ditto for the sine. In the current problem the latter is much preferred because it is the only one in which you can use $f(x) = x^2$.

And looking ahead, I'm guessing you might want to use the even periodic extension of x², which would be $4\pi$ periodic. As I mentioned before, the usual series for this is the periodic extension of x² on $(-\pi,\pi)$, but that is apparently not what you have been asked to do.

So I use the even periodic extension for f(x): that is, f(x) = x² for -2$$\pi$$<x<2$$\pi$$, and then I make the following calculations?:

$${a_0} = \frac{1}{{2\pi }}\int\limits_{ - 2\pi }^{2\pi } {{x^2}dx}$$

$${a_n} = \frac{1}{{2\pi }}\int\limits_{ - 2\pi }^{2\pi } {{x^2}\cos (\frac{1}{{2\pi }}nx)dx}$$

$${b_n} = \frac{1}{{2\pi }}\int\limits_{ - 2\pi }^{2\pi } {{x^2}\sin (\frac{1}{{2\pi }}nx)dx}$$

Is that what you're saying?

 

[LCKurtz]

There shouldn't be any $\pi$ in your cosines and sines. $2p=4\pi$ so
$$\sin{\frac{n\pi x}{p}} = sin{\frac n 2 x}$$

 

[libelec]

But that's the way?

 

[LCKurtz]

There shouldn't be any $\pi$ in your cosines and sines. $2p=4\pi$ so
$$\sin{\frac{n\pi x}{p}} = sin{\frac n 2 x}$$

And, once more, I will add that I suspect you are doing the wrong problem. Maybe there is a typo somewhere. Because if you take the periodic extension of x² on $(-\pi,\pi)$ and evaluate its series at $x = \pi$ you will get your desired answer.

 

[libelec]

OK, thanks. One question, though, when do I know when to use the periodic extensions (even an odd extensions)?

 

[LCKurtz]

OK, thanks. One question, though, when do I know when to use the periodic extensions (even an odd extensions)?

That's a good question. It depends on what you intend to do with the series, but generally, the smoother the periodic function is, the faster the series will converge. In this case, if you take the even extension of x² you get a continuous function whereas if you take the odd extension you get jump discontinuities at the ends. And the even extension has no sine terms so that's half the work. Another hint in this problem would be the series you are asked about has terms like 1/n² in it which suggests you need a continuous periodic function.

 

[libelec]

Thank you.