Series of functions and power series

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Abstract

The idea of approximating a function by a sequence of simple functions, or known ones, lies at the core of several mathematical techniques, both theoretical and practical. For instance, to prove that a differential equation has a solution one can construct recursively a sequence of approximating functions and show they converge to the required solution. At the same time, explicitly finding the values of such a solution may not be possible, not even by analytical methods, so one idea is to adopt numerical methods instead, which can furnish approximating functions with a particularly simple form, like piecewise polynomials. It becomes thus crucial to be able to decide when a sequence of maps generates a limit function, what sort of convergence towards the limit we have, and which features of the functions in the sequence are inherited by the limit. All this will be the content of the first part of this chapter.

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Canuto, C., & Tabacco, A. (2015). Series of functions and power series. In UNITEXT - La Matematica per il 3 piu 2 (Vol. 85, pp. 33–74). Springer-Verlag Italia s.r.l. https://doi.org/10.1007/978-3-319-12757-6_2

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