Convergence of an Infinite Series

An infinite series \sum {u_n} is said to converge, diverge or oscillate according as 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 it is a rational, or is an 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}


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.

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