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.