Trigonometric Fourier series are, in general, difficult to sum to high accuracy. An example is given by the series urn:x-wiley:00222526:media:sapm1994923213:sapm1994923213-math-0001 in which α and β(>0) are rational numbers satisfying 0<β/α≤1, where λ is an independent variable and j is a positive integer or zero. This paper presents a method for the efficient evaluation of the sum of such series. Fourier series which are the real or the imaginary part of , but which are not explicitly expressible as simple polynomials in λ, are obtained as the sum of a logarithic term and an infinite series in powers of λ, whose expansion is valid when 0<λ≤(2π/α) and is exact. When the Fourier series is expressible as a polynomial in λ, the method identifies that polynomial.