# 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.