Infinite expression

In mathematics, an infinite expression is a mathematical expression in which some operators take an infinite number of arguments, or in which the nesting of the operators continues to an infinite depth.

Examples of infinite expressions include infinite sums, whether expressed using summation notation or as a infinite series, such as

$\sum _{n=0}^{\infty }a_{n}=a_{0}+a_{1}+a_{2}+\ldots \,;$

infinite products, whether expressed using product notation or expanded as

$\prod _{n=0}^{\infty }b_{n}=b_{0}\times b_{1}\times b_{2}\times \ldots \,,$

and infinite continued fractions, such as

$[c_{0};c_{1},c_{2},c_{3},c_{4},\ldots ]=c_{0}+{\cfrac {1}{c_{1}+{\cfrac {1}{c_{2}+{\cfrac {1}{c_{3}+{\cfrac {1}{c_{4}+\ddots }}}}}}}}\,.$

For some infinite expresssions, the value may be ambiguous or not well defined; for instance, there are multiple summation rules available for assigning values to divergent series, and the same series may have different values according to different summation rules.