Mathematical Analysis

Sequences and Series

What are the concepts of convergent and divergent sequences?

Convergent and divergent sequences are based on the limit of a sequence. A convergent sequence, the one most commonly worked on in calculus, means that one mathematical sequence gets close to another and eventually approaches a limit (convergence can also apply to curves, functions, or series). This is seen visually when a curve approaches the x or y axes but does not quite reach it. For example, take the sequence of numbers used above, or {xn}n≥1. Often the numbers come closer and closer to a number we’ll call L; written in calculus, xn ≈ L. If the numbers do come closer, the sequence is said to be convergent and has a limit equal to L. Conversely, if the sequence is not convergent, it is called divergent.

Most mathematicians and scientists are not only interested in how a sequence converges (or diverges), but also how fast it converges, which is called the speed of convergence. There are several basic properties of the limits of a sequence, including that the limit of a convergent sequence is unique, every convergent sequence is bounded, and any bounded increasing or decreasing sequence is convergent.


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