How Do Fourier Series Solve Problems and Explore Their Applications?

AI Thread Summary
Fourier series are used to solve problems by decomposing periodic functions into sums of sine and cosine functions, which helps in analyzing complex waveforms. The discussion highlights confusion regarding the substitution of 2n-1 for n in a specific problem, emphasizing that using odd integers ensures certain properties of the cosine function at specific values of x. The rationale behind this substitution is to maintain the behavior of the function at critical points, such as x=1, where the cosine function yields consistent results. Additionally, there is a query about how to approach Fourier series within the context of precalculus, indicating a need for foundational understanding. Overall, the conversation revolves around clarifying the application of Fourier series and the significance of choosing odd integers in specific scenarios.
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What I know is that insert 2n-1 instead of n
please... explain to me
I don't know this reason
 

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Please make an effort to provide a more complete statement of your problem.
 
First I'm sorry that I can't write english well.
I have a problem's solution.
But I don't understand solution
For example Another problem's solution is n instead of 2n-1
But this problem's solution uses 2n-1 instead of n
I understood that 2n-1 is odd
but why do 2n-1 instead of n ?
 
When x= 1 or -1, 1-|x|= 0. Suppose you had just \frac{n\pi}{2}x. Then for even values of n, x= 1, you would have cos(m\pi) (m= n/2) which is either -1 or 1. By restricting the numerator to be odd, you are making sure that you always have, for x= 1, cos(\frac{\pi}{2}), cos(\frac{3\pi}{2}), etc.

By the way, how does one do Fourier Series in precalculus?
 
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