5. Sequences and Series

The concept of a limiting process at infinity is one of the central ideas of mathematical analysis. It forms the basis for all its essential concepts, like continuity, differentiability, series expansions of functions, integration, etc. The transition from the discrete to the continuous constitutes the modelling strength of mathematical analysis. Discrete models of physical, technical or economic processes can often be better and easily understood, provided that the number of their atoms—their discrete building blocks—is sufficiently big, if they are approximated by a continuous model with the help of a limiting process. The transition from difference equations for biological growth processes in discrete time to differential equations in continuous time or the description of share prices by stochastic processes in continuous time are examples for that. The majority of models in physics are field models, that is, they are expressed in a continuous space and time structure. Even though the models are discretised again in numerical approximations, the continuous model is still helpful as a background, for example for the derivation of error estimates.