The first purpose of this paper is to describe different series which converge at varying speeds to Euler's constant, 7. Limits are set for the errors of approximation in these series. Our second aim is to illustrate the scope for numerical exploration of these and similar results by means of graphical calculators, spreadsheets or other computer programs. Such facilities are now increasingly easy to use and widely available to mathematics students. Exercises or project work based on this material serve to motivate the ideas of limits, size of error, order of convergence and methods of numerical approximation, and are a valuable means of developing students’ skills of investigation and analysis.