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
\[ \lim_{n\to\infty } n = \infty\text.\]
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.
