# Convergence of an Infinite Series

An infinite series $\sum {u_n}$ is said to converge, diverge or oscillate according to how its sequence of partial sums $\left\langle {{s_n}} \right\rangle$ converges, diverges or oscillates. In case $\left\langle {{s_n}} \right\rangle$ converges to s, then s is called the sum of the series $\sum {u_n}$ and we shall write $s = \sum {u_n}$ or

$s = {u_1} + {u_2} + \cdots + {u_n} + \cdots$

In this series

$\lim {\left( {1 + \frac{1}{n}} \right)^n} = \lim \left( {1 + \frac{1}{{1!}} + \frac{1}{{2!}} + \cdots + \frac{1}{{n!}}} \right)$

shall be written as

$\lim {\left( {1 + \frac{1}{n}} \right)^n} = 1 + \frac{1}{{1!}} + \frac{1}{{2!}} + \cdots + \frac{1}{{n!}} + \cdots$

Thus the infinite series

$1 + \frac{1}{{1!}} + \frac{1}{{2!}} + \cdots + \frac{1}{{n!}} + \cdots$

shall be denoted by e, i.e.

$e = 1 + \frac{1}{{1!}} + \frac{1}{{2!}} + \cdots + \frac{1}{{n!}} + \cdots$

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.

Example:

The exponential number “e” defined by $e = 1 + \frac{1}{{1!}} + \frac{1}{{2!}} + \cdots$ is an irrational number.

Solution:

Let e be a rational number and $e = \frac{m}{n}$, where m and n are natural numbers, then
$e = 1 + \frac{1}{{1!}} + \frac{1}{{2!}} + \cdots$
$\Rightarrow \frac{m}{n} – \left( {1 + \frac{1}{{1!}} + \frac{1}{{2!}} + \cdots + \frac{1}{{n!}}} \right) = \frac{1}{{\left( {n + 1} \right)!}} + \frac{1}{{\left( {n + 2} \right)!}} + \cdots$
$\Rightarrow n!\left\{ {\frac{m}{n} – \left( {1 + \frac{1}{{1!}} + \frac{1}{{2!}} + \cdots + \frac{1}{{n!}}} \right)} \right\}$
$< \frac{1}{{\left( {n + 1} \right)}} + \frac{1}{{{{\left( {n + 1} \right)}^2}}} + \frac{1}{{{{\left( {n + 1} \right)}^3}}} + \cdots = \frac{{\frac{1}{{n + 1}}}}{{1 – \frac{1}{{n + 1}}}} = \frac{1}{n}$

is a contradiction as no natural number is less than $\frac{1}{n}{\text{ }}\forall n \in \mathbb{N}$.

Hence, e is an irrational number.

The fact that $\sum {u_n}$ converges to s shall also be expressed as $\sum {u_n} \to s$. Similarly we shall write $\sum {u_n} \to + \infty$ or$– \infty$ according as the series $\sum {u_n}$ diverges to$+ \infty$. If a series is finite, say, consisting of only m terms, then $\left\langle {{s_n}} \right\rangle \to {s_m}$. Thus every finite series is a convergent series, as such investigations regarding convergence are required for infinite series only.