# What are two examples of divergent sequences?

Jul 1, 2015

${U}_{n} = n$ and ${V}_{n} = {\left(- 1\right)}^{n}$

#### Explanation:

Any series that is not convergent is said to be divergent

${U}_{n} = n$ :

${\left({U}_{n}\right)}_{n \in \mathbb{N}}$ diverges because it increases, and it doesn't admit a maximum :

${\lim}_{n \to + \infty} {U}_{n} = + \infty$

${V}_{n} = {\left(- 1\right)}^{n}$ :

This sequence diverges whereas the sequence is bounded :
$- 1 \le {V}_{n} \le 1$

Why ?

A sequence converges if it has a limit, single !

And ${V}_{n}$ can be decompose in 2 sub-sequences :

${V}_{2 n} = {\left(- 1\right)}^{2 n} = 1$ and
${V}_{2 n + 1} = {\left(- 1\right)}^{2 n + 1} = 1 \cdot \left(- 1\right) = - 1$

Then : ${\lim}_{n \to + \infty} {V}_{2 n} = 1$
${\lim}_{n \to + \infty} {V}_{2 n + 1} = - 1$

A sequence converges if and only if every sub-sequences converges to the same limit.

But ${\lim}_{n \to + \infty} {V}_{2 n} \ne {\lim}_{n \to + \infty} {V}_{2 n + 1}$

Therefore ${V}_{n}$ doesn't have a limit and so, diverges.