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. A generic concept for infinite expression can lead to ill-defined or self-inconsistent constructions, but there are several instances of infinite expressions that are well-defined.

Examples

Examples of well-defined infinite expressions are
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 is obtained in every case from the sequence of finite expressions, by evaluating the sequence at a hypernatural value of the index n, and applying the standard part, so that.