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LIMITS CONVERGENCE
| Content Provider | Scilit |
|---|---|
| Author | Apelian, Christopher Surace, Steve |
| Copyright Year | 2009 |
| Description | One could argue that the concept of limit is themost fundamental one in analysis. Two of the most important operations in a first-year calculus course, the derivative and the integral, are in fact defined in terms of limits, even though many first-year calculus students willingly forget this is so. We begin this chapter by considering limits of sequences. The limit of a sequence of numbers, while the simplest kind of limit, is really just a special case of the limit of a sequence of vectors. In fact, real numbers can be considered geometrically as points in one-dimensional space, while vectors with k real components are just points in k-dimensional space. The special case of k = 2 corresponds to limits of sequences of points in R2 and to limits of sequences of points in C. Whether a sequence of real numbers, a sequence of real vectors, or a sequence of complex numbers has a well-defined limit is just a matter of determining whether the sequence of points is converging in some sense to a unique point in the associated space. This notion of convergence is one of the distance-related concepts referred to in the previous chapter, and it is common to all the spaces of interest to us. For this reason, we will again use the symbol X to denote any of the spaces R, Rk, or C in those cases where the results apply to all of them. After establishing the ideas underlying convergence of sequences in X, we develop the related notion of a series, whereby the terms of a sequence are added together. As we will see, whether a series converges to a well-defined sum depends on the behavior of its associated sequence of partial sums. While this definition of convergence for a series is both efficient and theoretically valuable, we will also develop tests for convergence that in many cases are easier to apply. Book Name: Real and Complex Analysis |
| Related Links | https://content.taylorfrancis.com/books/download?dac=C2009-0-00204-2&isbn=9780429142017&doi=10.1201/b12330-15&format=pdf |
| Ending Page | 103 |
| Page Count | 1 |
| Starting Page | 103 |
| DOI | 10.1201/b12330-15 |
| Language | English |
| Publisher | Informa UK Limited |
| Publisher Date | 2009-12-08 |
| Access Restriction | Open |
| Subject Keyword | Book Name: Real and Complex Analysis History and Philosophy of Science Behavior Convergence Calculus Defined Limits of Sequences Notion |
| Content Type | Text |
| Resource Type | Chapter |
