Infinite expression (mathematics)

In mathematics, an infinite expression is an expression in which some operators take an infinite number of arguments, or in which the nesting of the operators continues to an infinite depth.[1] A generic concept for infinite expression can lead to ill-defined or self-inconsistent constructions (much like a set of all sets), but there are several instances of infinite expressions that are well defined.

Examples of well-defined infinite expressions include[2][3] infinite sums, whether expressed using summation notation or as an infinite series, such as

\( \sum_{n=0}^\infty a_n = a_0 + a_1 + a_2 + \cdots \,; \)

infinite products, whether expressed using product notation or expanded, such as

\( \prod_{n=0}^\infty b_n = b_0 \times b_1 \times b_2 \times \cdots \, \)

infinite nested radicals, such as

\( \sqrt{1+2\sqrt{1+3 \sqrt{1+\cdots}}} \, \)

infinite power towers, such as

\( \sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\cdot^{\cdot^{\cdot}}}}} \, \)

and infinite continued fractions, whether expressed using Gauss's Kettenbruch notation or expanded, such as

\( c_0 + \underset{n=1}{\overset{\infty}{\mathrm K}} \, \frac{1}{c_n} = c_0 + \cfrac{1}{c_1 + \cfrac{1}{c_2 + \cfrac{1}{c_3 + \cfrac{1}{c_4 + \ddots}}}}\, \)

In infinitary logic, 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.
From the hyperreal viewpoint

From the point of view of the hyperreals, such an infinite expression E_\infty is obtained in every case from the sequence \langle E_n : n \in \mathbb{N}\rangle of finite expressions, by evaluating the sequence at a hypernatural value n=H of the index n, and applying the standard part, so that E_\infty=\text{st}(E_H).
See also

Iterated binary operation
Iterated function
Iteration
Dynamical system
Infinite word
Sequence
Decimal expansion
Power series
Analytic function
Quasi-analytic function

References

Helmer, Olaf (January 1938). "The syntax of a language with infinite expressions". Bulletin of the American Mathematical Society (Abstract) 44 (1): 33–34. doi:10.1090/S0002-9904-1938-06672-4. ISSN 0002-9904. OCLC 5797393..
Euler, Leonhard (November 1, 1988). Introduction to Analysis of the Infinite, Book I (Hardcover). J.D. Blanton (translator). Springer Verlag. p. 303. ISBN 978-0-387-96824-7.
Wall, Hubert Stanley (March 28, 2000). Analytic Theory of Continued Fractions (Hardcover). American Mathematical Society. p. 14. ISBN 978-0-8218-2106-0.