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 (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
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 formula_6 is obtained in every case from the sequence formula_7 of finite expressions, by evaluating the sequence at a hypernatural value formula_8 of the index "n", and applying the standard part, so that formula_9.