séquences

A sequence in mathematics is an ordered list of mathematical objects, usually numbers. These objects are called terms, and they are typically denoted by indices, such as $a_1, a_2, a_3, \dots$. The order of the terms is crucial, as changing the order can result in a different sequence.

Sequences can be finite, meaning they have a limited number of terms, or infinite, continuing without end. They can be defined by a specific formula, a rule, or by listing the terms. For example, the sequence of even numbers can be represented by the formula $a_n = 2n$ for $n \ge 1$, yielding $2, 4, 6, 8, \dots$. Another example is the Fibonacci sequence, where each term is the sum of the two preceding ones, starting from 0 and 1: $0, 1, 1, 2, 3, 5, 8, \dots$.

The study of sequences involves analyzing their properties, such as convergence (whether the terms approach a specific value as the index increases), divergence, monotonicity (whether the terms are always increasing or decreasing), and boundedness. Limits are a fundamental concept when discussing infinite sequences.

Sequences have wide-ranging applications in various fields, including computer science (algorithms, data structures), physics (modeling physical phenomena), economics (time series analysis), and statistics. They are a foundational concept in calculus and discrete mathematics.