Sequences

Definition:

A sequence is a function whose domain is of the form \(\{n\in\mathbb{z}: n \geq m\in\mathbb{Z} \}\). We often introduce sequences using the notation

\begin{align*} \{ a_m, a_{m+1}&, a_{m+2},\ldots \}\\ &\text{or}\\ \{&a_n\}_{n=m}^\infty \end{align*}

Although \(m\) is usually chosen to be \(0\) or \(1\), we often drop the indices and just use the notation \(\{a_n\}\), especially when the domain is understood or not terribly important.

Examples of Sequences

The Limit of a Sequence

Definition:

A sequence \(\{a_n\}\) of real numbers converges to a real number \(L\) provided that

for every \(\varepsilon > 0\) there is a natural number \(N\) such that \(n \geq N\) implies \(|a_n - L| < \varepsilon\).

If \(\{a_n\}\) converges to \(L\), we call the sequence convergent and write \(\lim_{n\to\infty} a_n = L\) or \(a_n \to L\) (as \(n\to\infty\)). \(L\) is called the limit of the sequence \(\{a_n\}\). A sequence that does not converge to some real number is said to diverge.