Fourier series and calculating a sum

In summary, the conversation discusses solving for the sum of k from 1 to infinity, 1/k^2 in the context of studying for an exam. The conversation also mentions using the function f(t) = pi + sum n from 1 to infinity: 2*(-1)^n/n*sin(nt) and rearranging it to calculate the sum. The idea of exploiting Parseval's identity is also brought up.
  • #1
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Homework Statement



Studying for an exam, this is a question from an earlier exam:

Calculate: sum k from 1 to infinity, 1/k^2

I have from previous question:
f(t)=pi+sum n from 1 to infinity: 2*(-1)^n/n*sin(nt)

f(t)=pi-t, -pi<=t<pi

So i need to rearrange this to calc the sum.


Homework Equations





The Attempt at a Solution



calc f(t) in a good point, I choose pi/2, use the left and right limit to get pi/2

then try top rearrange the series.


Especially I want to know mhow to do Sin(n*pi/2). it makes every second term disappear in the series but I don't know how to do this right. I tried replacing with n=2k+1 but it doesn't work.
 
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  • #2
The idea behind this kind of approach is to expolit Parseval's identity.
If f(x) can be expressed in terms of a Fourier series with coefficients [itex]a_n[/itex], then you have
[tex] \int_{-\pi}^\pi |f(x)|^2 dx = \sum_{n=-\infty}^\infty |a_n|^2 [/tex]
Can you find your original series somewhere in what you have obtained through the previous question?
 

1. What is a Fourier series?

A Fourier series is a mathematical representation of a periodic function as a sum of sine and cosine functions. It is used to approximate a complex function by breaking it down into simpler components.

2. How is a Fourier series calculated?

A Fourier series can be calculated using a formula that involves the coefficients of the sine and cosine functions. These coefficients can be determined by integrating the function over one period and solving for the coefficients using orthogonality properties.

3. What is the purpose of calculating a Fourier series?

The purpose of calculating a Fourier series is to approximate a complex function with simpler components, making it easier to analyze and understand. It is also used in many areas of science and engineering, such as signal processing and solving differential equations.

4. Can a Fourier series be used to represent any function?

No, a Fourier series can only represent periodic functions. If a function is not periodic, it cannot be represented by a Fourier series.

5. Are there any limitations to using Fourier series?

One limitation of using Fourier series is that it can only approximate a function, not represent it exactly. The accuracy of the approximation depends on the number of terms used in the series. Additionally, Fourier series may not converge for certain types of functions, such as piecewise continuous functions.

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