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
infinite sums, such as
∑ n = 0 ∞ a n = a 0 + a 1 + a 2 + ⋯ {\displaystyle \sum _{n=0}^{\infty }a_{n}=a_{0}+a_{1}+a_{2}+\cdots \,}
infinite products, such as
∏ n = 0 ∞ b n = b 0 × b 1 × b 2 × ⋯ {\displaystyle \prod _{n=0}^{\infty }b_{n}=b_{0}\times b_{1}\times b_{2}\times \cdots }
infinite nested radicals, such as
1 + 2 1 + 3 1 + ⋯ {\displaystyle {\sqrt {1+2{\sqrt {1+3{\sqrt {1+\cdots }}}}}}}
infinite power towers, such as
2 2 2 ⋅ ⋅ ⋅ {\displaystyle {\sqrt {2}}^{{\sqrt {2}}^{{\sqrt {2}}^{\cdot ^{\cdot ^{\cdot }}}}}}
infinite continued fractions, such as
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