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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, an infinite expression is an expression in which some operators take an infinite number of
arguments An argument is a series of sentences, statements, or propositions some of which are called premises and one is the conclusion. The purpose of an argument is to give reasons for one's conclusion via justification, explanation, and/or persua ...
, or in which the nesting of the operators continues to an infinite depth. A generic concept for infinite expression can lead to ill-defined or self-inconsistent constructions (much like a
set of all sets In set theory, a universal set is a set which contains all objects, including itself. In set theory as usually formulated, it can be proven in multiple ways that a universal set does not exist. However, some non-standard variants of set theory inc ...
), but there are several instances of infinite expressions that are well-defined.


Examples

Examples of well-defined infinite expressions are *
infinite sum In mathematics, a series is, roughly speaking, an addition of infinitely many terms, one after the other. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathemati ...
s, such as :: \sum_^\infty a_n = a_0 + a_1 + a_2 + \cdots \, *
infinite product In mathematics, for a sequence of complex numbers ''a''1, ''a''2, ''a''3, ... the infinite product : \prod_^ a_n = a_1 a_2 a_3 \cdots is defined to be the limit of the partial products ''a''1''a''2...''a'n'' as ''n'' increases without bound ...
s, such as :: \prod_^\infty b_n = b_0 \times b_1 \times b_2 \times \cdots * infinite nested radicals, such as :: \sqrt * infinite power towers, such as :: \sqrt^ * infinite continued fractions, such as :: c_0 + \underset \frac = c_0 + \cfrac, : where the left hand side uses
Gauss Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, Geodesy, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observat ...
's Kettenbruch notation. In
infinitary logic An infinitary logic is a logic that allows infinitely long statements and/or infinitely long proofs. The concept was introduced by Zermelo in the 1930s. Some infinitary logics may have different properties from those of standard first-order lo ...
, one can use infinite conjunctions and infinite disjunctions. Even for well-defined infinite expressions, the ''value'' of the infinite expression may be ambiguous or not well-defined; for instance, there are multiple summation rules available for assigning values to series, and the same series may have different values according to different summation rules if the series is not
absolutely convergent In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series \textstyle\sum_^\infty a_n is said ...
.


See also

*
Iterated binary operation In mathematics, an iterated binary operation is an extension of a binary operation on a set ''S'' to a function on finite sequences of elements of ''S'' through repeated application. Common examples include the extension of the addition operation ...
* Infinite word *
Decimal expansion A decimal representation of a non-negative real number is its expression as a sequence of symbols consisting of decimal digits traditionally written with a single separator: r = b_k b_\cdots b_0.a_1a_2\cdots Here is the decimal separator ...
*
Power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...
*
Infinite compositions of analytic functions In mathematics, infinite Function composition, compositions of analytic functions (ICAF) offer alternative formulations of Generalized continued fraction, analytic continued fractions, series (mathematics), series, product (mathematics), products ...
*
Omega language In formal language theory within theoretical computer science, an infinite word is an infinite-length sequence (specifically, an ω-length sequence) of symbols, and an ω-language is a set of infinite words. Here, ω refers to the first infinite o ...


References

{{Reflist Abstract algebra Mathematical analysis