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 are^[2]

• infinite sums, such as

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

• infinite products, such as

${\displaystyle \prod _{n=0}^{\infty }b_{n}=b_{0}\times b_{1}\times b_{2}\times \cdots }$

• infinite nested radicals, such as

${\displaystyle {\sqrt {1+2{\sqrt {1+3{\sqrt {1+\cdots }}}}}}}$

• infinite power towers,^[3] such as

${\displaystyle {\sqrt {2}}^{{\sqrt {2}}^{{\sqrt {2}}^{\cdot ^{\cdot ^{\cdot }}}}}}$

• infinite continued fractions, such as

${\displaystyle c_{0}+{\underset {n=1}{\overset {\infty }{\mathrm {K} }}}{\frac {1}{c_{n}}}=c_{0}+{\cfrac {1}{c_{1}+{\cfrac {1}{c_{2}+{\cfrac {1}{c_{3}+{\cfrac {1}{c_{4}+\ddots }}}}}}}},}$

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 point of view of the hyperreal numbers, such an infinite expression ${\displaystyle E_{\infty }}$ is obtained in every case from the sequence ${\displaystyle (E_{n}:n\in \mathbb {N} )}$ of finite expressions, by evaluating the sequence at a hypernatural value ${\displaystyle n=H}$ of the index n, and applying the standard part, so that ${\displaystyle E_{\infty }=\operatorname {st} (E_{H})}$.^{[citation needed]}

• Iterated binary operation
• Infinite word
• Decimal expansion
• Power series
• Infinite compositions of analytic functions
• Omega language