Infinite expression
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
Examples of well-defined infinite expressions are[2]
- infinite sums, such as
- [math]\displaystyle{ \sum_{n=0}^\infty a_n = a_0 + a_1 + a_2 + \cdots \, }[/math]
- infinite products, such as
- [math]\displaystyle{ \prod_{n=0}^\infty b_n = b_0 \times b_1 \times b_2 \times \cdots }[/math]
- infinite nested radicals, such as
- [math]\displaystyle{ \sqrt{1+2\sqrt{1+3 \sqrt{1+\cdots}}} }[/math]
- infinite power towers,[3] such as
- [math]\displaystyle{ \sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\cdot^{\cdot^{\cdot}}}}} }[/math]
- infinite continued fractions, such as
- [math]\displaystyle{ 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}}}}, }[/math]
- where the left hand side uses Gauss' Kettenbruch notation.[4]
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 hyperreal numbers, such an infinite expression [math]\displaystyle{ E_\infty }[/math] is obtained in every case from the sequence [math]\displaystyle{ (E_n : n \in \mathbb{N}) }[/math] of finite expressions, by evaluating the sequence at a hypernatural value [math]\displaystyle{ n = H }[/math] of the index n, and applying the standard part, so that [math]\displaystyle{ E_\infty = \operatorname{st}(E_H) }[/math].[citation needed]
See also
- Iterated binary operation
- Infinite word
- Decimal expansion
- Power series
- Infinite compositions of analytic functions
- Omega language
References
- ↑ Helmer, Olaf (January 1938). "The syntax of a language with infinite expressions". Bulletin of the American Mathematical Society 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.
- ↑ Moroni, Luca (2019). "The strange properties of the infinite power tower". arXiv:1908.05559 [math.HO].
- ↑ Wall, Hubert Stanley (March 28, 2000). Analytic Theory of Continued Fractions (Hardcover). American Mathematical Society. p. 14. ISBN 978-0-8218-2106-0.
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