Infinity symbol
Mathematics

In mathematics, the infinity symbol is typically used to represent a potential infinity. For instance, in mathematical expressions with summations and limits such as

∑ n = 0 ∞ 1 2 n = lim x → ∞ 2 x − 1 2 x − 1 = 2, {\displaystyle \sum _{n=0}^{\infty }{\frac {1}{2^{n}}}=\lim _{x\to \infty }{\frac {2^{x}-1}{2^{x-1}}}=2,}

the infinity sign is conventionally interpreted as meaning that the variable grows arbitrarily large towards infinity, rather than actually taking an infinite value, although other interpretations are possible.

When quantifying actual infinity, infinite entities taken as objects per se, other notations are typically used. For example, ℵ 0 {\displaystyle \aleph _{0}} (aleph-nought) denotes the cardinal number representing the size of the set of natural numbers, and ω {\displaystyle \omega } (omega) denotes the smallest ordinal number which is larger than all natural numbers.

The infinity symbol may also be used to represent a point at infinity, especially when there is only one such point under consideration. This usage includes, in particular, the infinite point of a projective line, and the point added to a topological space to form its one-point compactification.
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