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- W4382808084 abstract "Sequences and series constitute the training ground in which the notions of convergence and limit are first discussed. To better exploit the general understanding of topological and metric spaces built in the previous chapters, we will organise their study in a hierarchical fashion. We will start discussing purely topological properties of sequences, like the definition of convergence itself; the uniqueness of the limit and the limit of subsequences. Then, we will discover what is added if a metric structure is assumed on the space in which the sequence is defined. We will review notions like the boundedness of converging sequences; the existence of the limit for Cauchy sequences and complete metric spaces. A series of results specific to $$mathbb {R}$$ will follow. Many of them derive from the complete order relation defined on this set. I will introduce the notion of upper and lower limits and the (partial) ordering of infinite and infinitesimal sequences. We will discover how to “average” sequences and review several convergence criteria for specific types of sequences. A brief review of the main results in normed linear spaces, essentially $$mathbb {R}^n$$ , is also presented. The discussion of series and the convergence of their associated partial sums is presented in $$mathbb {R}$$ because little is gained from considering more general structures. We will deal with the important harmonic and power series. I will discuss the difficulties inherent in dealing with the notion of series and illustrate the most common criteria for deciding on their convergence. Finally, we will explore the notion of sequences of functions and the properties of uniform convergence." @default.
- W4382808084 created "2023-07-02" @default.
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- W4382808084 date "2023-01-01" @default.
- W4382808084 modified "2023-09-23" @default.
- W4382808084 title "Sequences and Series" @default.
- W4382808084 doi "https://doi.org/10.1007/978-3-031-30316-6_5" @default.
- W4382808084 hasPublicationYear "2023" @default.
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