How Does Energy Convergence Relate to Fourier Series Representation?

AI Thread Summary
The discussion focuses on understanding the relationship between energy in a function and its Fourier series representation, specifically using the example function y = 3t. The Fourier representation yields a DC average of 3π, with sine terms but no cosine terms, and the energy is calculated using Parseval's theorem. After applying 13 harmonics, 99% of the original function's energy is captured, but the user seeks clarity on determining the limiting value for convergence. They express uncertainty about whether this limiting value exists and if the energy difference would be zero at that point. The inquiry emphasizes the broader concept of convergence in infinite series, rather than just Fourier series.
Jag1972
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I have been trying to understand the Fourier series and the relationship between the energy in the original function and its Fourier representation. The example function: y = 3t has a period of 2∏. The Fourier coefficients are:


The Fourier representation has a dc average of 3∏, it has no cosine terms but does have sine terms with amplitude equal to -6/n.

Using Persavals thereom I can determine the energy in the Fourier series:

\frac{1}{\pi}\int^{2\pi}_{0} 3t^{2} dt = a0/2^{2} + \sum bn^{n}

After using about 13 harmonics I got it to 99% of the energy of the original function. I do not know how to get to a limiting value which I think is called convergence. A stable value reached, if there is no stable value then the function diverges. I know there are tests for convergence and divergence but these will not give actual limiting values. My question is that how does one know what the actual limiting value is orthis just something we have to reach ourselves, also would the energy difference be 0 at this limiting value. I hope it will be.
Thanks in advance.
 
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That is not an electrical for Fourier question. You are asking about convergence of any infinite series.
 
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