Sequences

Summary

A sequence $\{a_n\}$ is an ordered list of numbers indexed by integers $n$ . It converges if $a_n$ approaches a finite limit as $n\to \mathrm{\infty }$ . Series are built by summing sequence terms.

Prerequisites

Limits

Definition

A sequence is a function $n\mapsto a_n$ from $\mathbb{N}$ (or $\mathbb{N}\cup \{0\}$ ) to $\mathbb{R}$ . We write $\{a_n{\}}_{n=1}^{\mathrm{\infty }}$ .

means: for every $\epsilon >0$ there exists $N$ such that $n>N$ implies $|a_n-L|<\epsilon$ .

Types (examples)

• Arithmetic: $a_n=a+(n-1)d$ (e.g. $1,3,5,7,\dots$ ).
• Geometric: $a_n=a{r}^{n-1}$ (e.g. $2,6,18,54,\dots$ ).
• Recursive: Fibonacci $F_n=F_{n-1}+F_{n-2}$ .

Worked Example

$a_n=1/n\to 0$ (convergent).
$b_n=n\to \mathrm{\infty }$ (diverges).
$c_n=(-1{)}^n$ diverges by oscillation.

Common Mistakes

• Confusing the limit of a sequence $a_n$ with the sum of the series $\sum a_n$ .
• Claiming every bounded sequence converges (false in $\mathbb{R}$ without monotonicity; true for monotone bounded sequences).

Connections

• Partial sums of series are sequences: Series Sums by Partial Sums
• Next: Infinite Series

References

Sequences are introduced before series in OpenStax Calculus Volume 2.^[1]