The idea of convergence in mechanics

Many results of mechanical engineering (and other areas of engineering and mathematics) are approximate; they are calculated using general computer implementations of the finite element method. Some results take the form of series expansions. Sometimes the problem of convergence of such series or approximations can be studied using the simple tools of calculus. On other occasions the analysis can be quite complicated, requiring us to apply methods from functional analysis. We would like to shed some light on the nature of convergence, especially as it relates to the practical problems of mechanics. Of course, convergence in mechanics does not differ from convergence in pure mathematics; however, within the framework of mechanics we can draw on examples in order to illustrate the importance of the topic. We begin with definitions and simple examples.