# What is meant by a convergent sequence?

A sequence is said to be convergent if it's limit exists.

Else, it's said to be divergent.

It must be emphasized that if the limit of a sequence ${a}_{n}$ is infinite, that is ${\lim}_{n \to \infty} {a}_{n} = \infty$ or ${\lim}_{n \to \infty} {a}_{n} = - \infty$, the sequence is also said to be divergent.

A few examples of convergent sequences are:

• $\frac{1}{n}$, with ${\lim}_{n \to \infty} \frac{1}{n} = 0$
• The constant sequence $c$, with $c \in \mathbb{R}$ and ${\lim}_{n \to \infty} c = c$
• ${\left(1 + \frac{1}{n}\right)}^{n}$, with ${\lim}_{n \to \infty} {\left(1 + \frac{1}{n}\right)}^{n} = e$ where $e$ is the base of the natural logarithms (also called Euler's number).

Convergent sequences play a very big role in various fields of Mathematics, from estabilishing the foundations of calculus, to solving problems in Functional Analysis, to motivating the development of Toplogy.