Divergence

A sequence that does not converge is said to diverge.

The idea is that the definition of convergence says:

\forall ε > 0 \exists N \in N \forall n \geq N (| a_{n} - L | < ε)

as a result, the negation is:

\exists ε > 0 \forall N \in N \exists n \geq N (| a_{n} - L | \geq ε)

Note

What about $L$ ? For the definition of convergence we say that $L$ exists, so for divergence we must say that for any arbitrary $L$ we can choose our $ε$ .

Example

Why does the following sequence diverge?

(1, - 1, 1, - 1, 1, - 1, \dots)

Proof
Let $L \in R$ . Notice that since $a_{n} = (- 1)^{n}$ , then:

| a_{n} - L | = | (- 1)^{n} - L | = | 1 - L | \lor | 1 + L |

We have two cases:

$a_{n} = 1$ . Then choose $ε = \frac{1}{10}$ for example. Then we ask if $1 \in V_{1 / 10} (L)$ ? Namely is:

\frac{1}{10} - L < 1 < \frac{1}{10} + L

But if that's the case then notice that the next term for $a_{n}$ is -1 and $- 1 \notin V_{1 / 10} (L)$ is a contradiction.
2. $a_{n} = - 1$ follows reverse of our argument above.
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