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:(adsbygoogle = window.adsbygoogle || []).push({});

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:

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

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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# Energy of the Fourier Series

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