The rate of convergence of a Fourier‐series representation of a given function depends on the nature of the function and of its derivatives. This is shown by using the graphical outputs of a desk computer for different cases. For full‐range series, the effects of continuity and discontinuity of the function and its first derivatives are shown first. The advantage of half‐range formulae due to the free choice of function in the second half‐range are demonstrated next, along with the importance of choice of sine and cosine series according to the function being represented. Finally, a method is given of modifying the given function in a simple manner whereby a dramatic increase in the rate of convergence of the Fourier series is obtained. Examples of the Gibbs overshoot and its elimination are included.