# Convergence of an Infinite Series

An infinite series is said to converge, diverge or oscillate according as its sequence of partial sums converges, diverges or oscillates. In case converges to *s*, then *s* is called the **sum of the series ** and we shall write or

In this series

shall be written as

Thus the infinite series

shall be denoted by *e*, i.e.

By virtue of the nature of the terms of a convergent series we are sometimes able to ascertain the type of the value of the sum of the series such as it is a rational, or is an irrational number.

__Example__:

The exponential number “*e*” defined by is an irrational number.

__Solution__:

Let *e* be a rational number and , where *m* and *n* are natural numbers, then

a contradiction as no natural number is less than .

Hence, *e* is an irrational number.

The fact that converges to *s* shall also be expressed as . Similarly we shall write or according as the series diverges to. If a series is finite, say, consisting of only *m* terms then . Thus every finite series is a convergent series. As such investigations regarding convergence are required for infinite series only.