An infinite series is said to converge, diverge or oscillate according to how 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 whether it is a rational or is irrational number.
The exponential number “e” defined by is an irrational number.
Let e be a rational number and , where m and n are natural numbers, then
is 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.