Convergence of a Sequence

Convergence of a sequence

A Sequence $(a_{n})$ of real numbers converges to some $L \in R$ if:

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

Geometrically this is saying:

Given some error $ε$ , you can always choose a start point in the sequence $a_{n}$ at point $N$ such that all further terms are within the range of $a_{n} - L$ and $a_{n} + L$ .

We write:

L = lim_{n \to \infty} a_{n} = lim a_{n} \equiv a_{n} \to L

when $L$ is the limit of $(a_{n})$ .

If $∄ L$ then we say $(a_{n})$ diverges; namely if $(a_{n})$ doesn't converge then it diverges.