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These are sufficient conditions for a "well-behaved" function to have a convergent Fourier Series.

If f(x) is a bounded periodic function with a finite number of maxima, minima and discontinuities in 0<=x<T (where T is the period) then the Fourier series for f(x) converges to f(x) for all x at which f(x) is continuous. Where f(x) is not continuous, the series conveges to the midpoint of the discontinuity, that is: ½(f(x+) + f(x-))

Note that though these conditions are sufficient, they are not necessary. For example, sin(x-1) has a convergent Fourier Series.

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