Sequences

From arithmetic and geometric sequences to the $\Sigma$ notation

Overview

A sequence is an ordered list of numbers following a rule. In high school, one learns formulas for arithmetic and geometric sequences and how to compute sums using the $\Sigma$ notation; in university, these ideas extend to convergence and divergence.

$a_1$
$a_2$
$a_3$
$\cdots$
$a_n$
$\cdots$

Figure 1: A sequence $\{a_n\}$ — an ordered list of numbers following a rule.

Table of Contents by Level

Introduction

Arithmetic sequences, geometric sequences, $\Sigma$ notation and formulas. High school level.

Main Topics

  • Arithmetic sequence: a sequence in which the difference between consecutive terms is constant.
  • Geometric sequence: a sequence in which the ratio between consecutive terms is constant.
  • $\Sigma$ notation: a symbol for sums, with its basic formulas.
  • Recurrence relation: an equation that defines a sequence inductively.

Frequently Asked Questions

What is a mathematical sequence?

A sequence is an ordered list of numbers $a_1, a_2, a_3, \ldots$ indexed by natural numbers ($n=1,2,3,\ldots$). It can be viewed as a function $\mathbb{N}\to\mathbb{R}$ (or $\mathbb{C}$). For infinite sequences, the key concepts are limits, convergence, and divergence.

What does it mean for a sequence to converge?

$\lim_{n\to\infty} a_n = L$ means that $a_n$ gets arbitrarily close to $L$ as $n$ grows. Formally: for every $\varepsilon>0$, there exists $N$ such that $n>N$ implies $|a_n-L|<\varepsilon$. If such a finite $L$ exists, the sequence is said to converge to $L$.

How does studying sequences help in university mathematics?

Sequence limits are the foundation of calculus definitions (the $\varepsilon$-$N$ definition of limits). They are essential in numerical analysis (convergence of iterative methods), probability (law of large numbers), Fourier analysis (Fourier coefficients), and complex analysis (radius of convergence of power series).