Sequences and series

thumb|A part of an infinite sequence of real numbers (in blue), indexed by a natural number . This sequence is neither increasing, decreasing, convergent, nor Cauchy. It is, however, bounded (by red dashed lines). In mathematics, a sequence is a collection of objects possibly with repetition, that come in a specified order. Like a set, it contains members (also called elements, or terms). Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. The notion of a sequence can be generalized to an indexed family, defin

sequence of numbers with constant differences between consecutive numbers

sequence of numbers where each term is found by multiplying the previous one by a fixed, non-zero number

null sequence; value that the terms of a sequence "tend to"

sequence whose elements become arbitrarily close to each other

In mathematics, a tuple is a finite sequence (or ordered list) of numbers. More generally, it is a sequence of mathematical objects, called the elements of the tuple. An -tuple is a tuple of elements, where is a non-negative integer. There is only one 0-tuple, called the empty tuple. A 1-tuple and a 2-tuple are commonly called a singleton and an ordered pair, respectively. The term "infinite tuple" is occasionally used for "infinite sequences".

In mathematics, a subsequence of a given sequence is a sequence that can be derived from the given sequence by deleting some or no elements without changing the order of the remaining elements. For example, the sequence \langle A,B,D \rangle is a subsequence of \langle A,B,C,D,E,F \rangle obtained after removal of elements C, E, and F. The relation of one sequence being the subsequence of another is a partial order.

expression representing the product of an infinite sequence

progression formed by taking the reciprocals of an arithmetic progression

concept in mathematics

'best' approximation of a function by a rational function of given order

increasing sequence of reduced fractions between 0 and 1 whose denominators do not exceed a given positive integer

decimal number formed by concatenating successive representations of integers

inequality relating two increasing sequences of numbers, or one decreasing and another increasing sequence

mathematical operation of composing a function with itself repeatedly

theorem of calculus

sequence of random variables

In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to the sum of the function's values at each element. There are numerous examples of subadditive functions in various areas of mathematics, particularly norms and square roots. Additive maps are special cases of subadditive functions.

endless sequence

a vector space whose elements are infinite sequences of real or complex numbers

sequence valued in polynomials

sequence for which the same terms are repeated over and over

example of the simplest one-dimensional low-discrepancy sequence over the unit interval; it is constructed by reversing the base-n representation of the sequence of natural numbers (1, 2, 3, …)

In mathematics, a function f is superadditive if f(x+y) \geq f(x) + f(y) for all x and y in the domain of f.

type of mathematical sequence

amount of variation between extrema of a function or sequence