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

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 $e = 1 + \frac{1}{{1!}} + \frac{1}{{2!}} + \cdots$ is an irrational number.

Solution:

Let e be a rational number and $e = \frac{m}{n}$, where m and n are natural numbers, then

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.