Estimating the sum of reciprocal powers using a given fourier series

In summary, the conversation discusses finding the sums of the Fourier series for f(x), which is defined by the absolute value of x. The first sum, involving only odd n integers, is shown to be equal to pi^2/8. The second sum, which includes both odd and even n integers, is approximated by combining the sums for the odd and even terms. The final solution for S, the sum over all n, is obtained by solving for S using the previous results.
  • #1
buffordboy23
548
2

Homework Statement



Let f(x) be defined by the following Fourier series for [tex] \left|x\right|[/tex]:

[tex] f(x) = \frac{\pi}{2} - \frac{4}{\pi}\sum_{1,3,...}\frac{cos\left(nx\right)}{n^{2}}[/tex]

Show that

[tex] \sum_{1,3,...}\frac{1}{n^{2}} = \frac{\pi^{2}}{8}[/tex]

and

[tex] \sum_{1,2,3,...}\frac{1}{n^{2}} = \frac{\pi^{2}}{6}[/tex]

The Attempt at a Solution



I was able to find the first sum by letting x = 0. I don't now how to approach the second part since the sum consists of the odd and even n integers, but the Fourier series is only comprised of the odd integers.
 
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  • #2
I think I figured it out, or at least obtained a really close approximation.

Multiplying 1/4 to both sides of

[tex] \sum_{1,3,...}\frac{1}{n^{2}} = \frac{\pi^{2}}{8} [/tex]

will yield the sum for the terms 2, 6, 10, ...

[tex] \sum_{2,6,10,...}\frac{1}{n^{2}} = \frac{\pi^{2}}{32} [/tex]

Now multiplying this sum by 1/4 gives the terms 4, 8, 12,...

[tex] \sum_{4,8,12,...}\frac{1}{n^{2}} = \frac{\pi^{2}}{128} [/tex]

Combining all of the sums together gives a close approximation:

[tex] \sum_{n}\frac{1}{n^{2}} = \frac{21\pi^{2}}{128} \approx \frac{\pi^{2}}{6} [/tex]
 
  • #3
If S is the sum over all n (even or odd) then (1/4)S is the sum over all evens. S-(1/4)S is then the sum over all odds which is pi^2/8. Solve for S.
 
  • #4
Dick said:
If S is the sum over all n (even or odd) then (1/4)S is the sum over all evens. S-(1/4)S is then the sum over all odds which is pi^2/8. Solve for S.

EDIT: I see now. Duh! Thanks.
 
Last edited:

1. What is the purpose of estimating the sum of reciprocal powers using a given Fourier series?

The purpose of estimating the sum of reciprocal powers using a given Fourier series is to approximate the value of a sum of infinite terms using a finite number of terms. This can be useful in various mathematical and scientific applications, such as in signal processing or data analysis.

2. How does a Fourier series represent the sum of reciprocal powers?

A Fourier series is a representation of a periodic function as a sum of trigonometric functions. By manipulating the coefficients of these trigonometric functions, it is possible to approximate the sum of reciprocal powers by including terms with different frequencies and amplitudes.

3. What is the significance of the reciprocal powers being used in this estimation?

The reciprocal powers used in this estimation play a crucial role in approximating the value of the sum. By choosing the appropriate reciprocal powers, it is possible to achieve a more accurate estimation of the sum compared to using only a few terms of the Fourier series.

4. Can this method be used to estimate the sum of any series?

No, this method is specifically designed for estimating the sum of reciprocal powers using a given Fourier series. Other methods, such as Taylor series or Riemann sums, may be more suitable for estimating the sum of other types of series.

5. How can I determine the accuracy of the estimated sum using this method?

The accuracy of the estimated sum can be determined by comparing it to the exact value of the sum, if known. Alternatively, the accuracy can also be evaluated by increasing the number of terms in the Fourier series and observing the change in the estimated sum. Generally, the more terms included, the more accurate the estimation will be.

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