Using a Fourier Cosine Series to evaluate a sum

In summary, a Fourier Cosine Series is a mathematical tool used to represent a periodic function as a sum of cosine functions. It is used to evaluate sums by approximating the original function with a truncated series of cosine functions. The advantages of using this series include simplifying complex functions, providing accurate approximations, and having applications in signal processing and data compression. However, it has limitations for functions with sharp discontinuities or rapidly changing values. A Fourier Cosine Series is a special case of a Fourier Series, where all the coefficients of the sine functions are equal to zero, and it is useful for evaluating even functions. Other types of Fourier Series can be used for evaluating odd functions.
  • #1
richyw
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Homework Statement



a) Show that the Fourier Cosine Series of [itex]f(x)=x,\quad 0\leq x<L[/itex] is
[tex]x ~ \frac{L}{2}-\frac{4 L}{\pi ^2}\left[\left(\frac{\pi x}{L}\right)+ \frac{\cos\left(\frac{3\pi x}{L}\right)}{3^2}+\frac{\cos\left(\frac{5 \pi x}{L}\right)}{5^2}+\dots\right][/tex]

b) use the above series to evaluate the sum[tex]1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...[/tex]

Homework Equations



Fourier Cosine Series General form

The Attempt at a Solution



So I have done part a, but I am lost on how to do part b. I don't understand how to get the even terms? Perhaps I need to differentiate the series, which would pull an n out, making the n's even.

To differentiate term by term a cosine series I need f'(x) to be piecewise continuous. Which I think it is.
 
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  • #2
Fixed LaTeX:

richyw said:

Homework Statement



a) Show that the Fourier Cosine Series of [itex]f(x)=x,\quad 0\leq x<L[/itex] is
[tex]x \sim \frac{L}{2}-\frac{4 L}{\pi ^2}\left[ \cos\left(\frac{\pi x}{L}\right)+ \frac{\cos\left(\frac{3\pi x}{L}\right)}{3^2}+\frac{\cos\left(\frac{5\pi x}{L}\right)}{5^2}+ ...\right][/tex]

b) use the above series to evaluate the sum[tex]1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...[/tex]

Homework Equations



Fourier Cosine Series General form

The Attempt at a Solution



So I have done part a, but I am lost on how to do part b. I don't understand how to get the even terms? Perhaps I need to differentiate the series, which would pull an n out, making the n's even.

To differentiate term by term a cosine series I need f'(x) to be piecewise continuous. Which I think it is.
 
  • #3
micromass said:
Fixed LaTeX:

thanks. I did too. I had to quickly reinstall LaTeXiT from macports :smile:
 
  • #4
micromass said:
Anyway, what you need to do is choose ##x## (and perhaps ##L##) suitably such that you can extract the series ##\sum \frac{1}{n^2}## out of there.

but my Fourier series only has the odd-n terms?
 
  • #5
I'll assume the Fourier expansion is correct.

What you need to do is to choose ##x## (and perhaps ##L##) wisely in order to extract an interesting series.

Now, also note the following:

Take

[tex]S = 1 + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \frac{1}{5^2} + ...[/tex]

then

[tex]S = \left(1 + \frac{1}{3^2} + \frac{1}{5^2} + ... \right) + \left(\frac{1}{2^2} + \frac{1}{4^2} + ...\right)[/tex]

But

[tex]\frac{1}{2^2} + \frac{1}{4^2} + \frac{1}{6^2} + ...= \frac{1}{4}\left(1 + \frac{1}{2^2}+ \frac{1}{3^2} + ...\right)= \frac{S}{4}[/tex]

So differentiating is unnecessary.
 
  • #6
ok my initial guess was wrong obviously. becuase it would not make the n's even, it would make them not squared...
 
  • #7
First think about, what you get, setting [itex]x=L[/itex]. Then you can think further about how to get the given series from this!
 
  • #8
ok. I don't really have time to show my work, but I ended up with π^2/6. Is this the correct answer?

the trick I didn't get was that part with the s/4
 
  • #9
richyw said:
ok. I don't really have time to show my work, but I ended up with π^2/6. Is this the correct answer?

the trick I didn't get was that part with the s/4

You have the correct answer!

http://en.wikipedia.org/wiki/Basel_problem
 
  • #10
micromass said:
Fixed LaTeX:

Actually, this is the Fourier series of the even function ##f(x) = |x|, -L \leq x \leq L##, extended to the whole real line as a periodic function of period ##2L##. Basically, the person posing the question needs to specify ##f(x)## outside the desired interval ##[0,L]##. If that is not done correctly the Fourier series won't be pure 'cosine' and/or might not be continuous at ##\pm \, L##, leading to a mis-match between f and the sum of the series at those points.
 

FAQ: Using a Fourier Cosine Series to evaluate a sum

1. What is a Fourier Cosine Series?

A Fourier Cosine Series is a mathematical tool used to represent a periodic function using a sum of cosine functions with different frequencies and amplitudes. It is a way to break down a function into simpler components for analysis and evaluation.

2. How is a Fourier Cosine Series used to evaluate a sum?

A Fourier Cosine Series is used to evaluate a sum by approximating the original function with a truncated series of cosine functions. The more terms included in the series, the closer the approximation will be to the original function. By using this series, we can evaluate a sum by finding the coefficients of the cosine functions and adding them together.

3. What are the advantages of using a Fourier Cosine Series to evaluate a sum?

One advantage is that it allows us to represent a complex function in a simpler form, making it easier to analyze and manipulate. Additionally, it can provide a more accurate approximation of a function compared to other methods, especially for periodic functions. It also has applications in signal processing and data compression.

4. Are there any limitations to using a Fourier Cosine Series to evaluate a sum?

Yes, there are some limitations. A Fourier Cosine Series can only be used for functions that are periodic and have a single-valued representation. It may also struggle with functions that have sharp discontinuities or rapidly changing values. In these cases, a different method may be more appropriate for evaluating a sum.

5. How is a Fourier Cosine Series related to other Fourier Series?

A Fourier Cosine Series is a special case of a Fourier Series, where all the coefficients of the sine functions are equal to zero. This means that the series will only contain cosine functions, making it useful for evaluating even functions. Other types of Fourier Series, such as Fourier Sine Series, can be used for evaluating odd functions.

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