Finding the Fourier cosine series for ##f(x)=x^2##

chwala
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Homework Statement
See attached.
Relevant Equations
Fourier cosine series
I was just going through my old notes on this i.e

1699522218550.png
The concept is straight forward- only challenge phew :cool: is the integration bit...took me round and round a little bit... that is for ##A_n## part.

My working pretty ok i.e we shall realize the text solution. Kindly find my own working below.

1699522329913.png
Now to my question supposing we have to find say Fourier cosine series for ##f(x)= x^7##. The integration by parts here will take like forever to do. Do we have software for this? Will Wolfram help?
 
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chwala said:
Will Wolfram help?
Pretty easy to check for yourself :wink: . Let us know !

##\ ##
 
BvU said:
Pretty easy to check for yourself :wink: . Let us know !

##\ ##
Will do check @BvU ... however, if one was to do this by hand ... No technology...how long would it take to work to solution?
 
The ##f(x)= x^7## is odd. Perhaps Fourier sine series rather than cosine?
 
chwala said:
Now to my question supposing we have to find say Fourier cosine series for ##f(x)= x^7##. The integration by parts here will take like forever to do. Do we have software for this? Will Wolfram help?

Set I_n = \int_0^{2\pi} x^{2n+1} \sin kx\,dx, \qquad k = 1, 2, \dots. so that integrating by parts twice, <br /> I_n = - \frac{(2n+1)(2n)}{k^2}I_{n-1} - \frac{(2\pi)^{2n+1}}{k}. This recurrence relation can be solved subject to the initial condition I_0 = -\frac{2\pi}{k} to obtain <br /> I_N = \frac{(2N + 1)!(-1)^N}{k^{2N}} \frac{2\pi}{k}\sum_{n=0}^N \frac{(-1)^{n+1}(2\pi k)^{2n}}{(2n+1)!}.
 
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There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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