Sequence

A sequence is a function whose domain is the natural numbers, so each natural number indexes a term. Its terms may repeat, and their order matters.

Function: A relation that assigns each element of a domain exactly one element of a codomain. A sequence is a function with natural-number indices.
Arithmetic progression: A sequence in which the difference between consecutive terms is constant. Adding a fixed difference at each step generates this basic sequence type.
Series (mathematics): A sum whose terms are taken from a sequence, often considered through its partial sums. A sequence supplies the terms whose infinite sum may converge or diverge.
Set (mathematics): A collection of distinct objects in which order does not matter. A sequence preserves order and permits repetitions, unlike a set.
Bolzano–Weierstrass theorem: The theorem that every bounded sequence in Euclidean space has a convergent subsequence. It guarantees convergent structure inside every bounded sequence of real vectors.
Natural number: A number used for counting, conventionally beginning at zero or one. Natural numbers serve as the positions that index sequence terms.
Geometric progression: A sequence in which each term after the first is obtained by multiplying the previous term by a constant ratio. Repeated multiplication generates terms and determines their growth or decay.
Taylor series: A power series whose coefficients are determined by derivatives of a function at a point. Its coefficients form a sequence used to approximate functions near that point.
Multiset: A collection in which elements may occur multiple times but order does not matter. It retains repetition while discarding the positional order of sequence terms.
Subsequence: A sequence formed by selecting terms at strictly increasing indices from another sequence. Subsequences reveal limiting behavior that the full sequence may obscure.

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