Convergence and divergence

Limits part 1: Infinite sequences: Basic calculation rules for limits

Convergence and divergence

In the previous section we discussed what is means for a sequence to diverge to infinity:

We say that a sequence $(a_n)_{n=1}^\infty$ diverges to infinity if for every $M$ there exists $N$ such that $a_n > M$ for all $n > N$ .

We looked at the sequence $1, 2, 3, 4, \dots$ and we claimed that

$\lim_{n\to\infty } n = \infty\text.$

Let $M$ be given and choose $N = M$ . For $n > N$ we indeed have $a_n = n > M$ as desired.

It makes sense to now look at the sequence $-1, -2, -3, -4, \dots$ . We want to express that given a constant this sequence becomes smaller than this constant after a while. Another way of saying this is that there is no lower bound $M$ such that all $a_n$ are at least $M$ . More precisely,

We say that a sequence $(a_n)_{n=1}^\infty$ diverges to minus infinity if for every $M$ there exists $N$ such that $a_n < M$ for all $n > N$ .

We write this as $\lim_{n\to\infty} a_n = -\infty$ .

There exist sequences that neither converge nor diverge to $\infty$ or $-\infty$ . In this animation you can see that the sequence

$0,1,0,1,0,1,0,1,\dots$ is bounded but does not converge.