What is Sequence?

A sequence $(u_n)$ assigns to each integer $n \geq 0$ (or $n \geq 1$) a real number $u_n$, its term of index $n$. A sequence can be given

• explicitly, by a formula for $u_n$ in terms of $n$: e.g. $u_n = n^2 + 1$;
• recursively, by its first term and a rule to pass from each term to the next: e.g. $u_0 = 3$ and $u_{n+1} = 2u_n - 1$.

A sequence is a function $u \colon \N \to \R$ (or from $\{n \in \N : n \geq n_0\}$ to $\R$). The image of $n$ is written $u_n$, and the sequence itself $(u_n)_{n\in\N}$ or simply $(u_n)$.