## 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 {*x*_{n}}_{n≥1}. Often the numbers come closer and closer to a number we’ll call *L*; written in calculus, * x_{n} ≈ 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.