**Example 1.** Vaughan gave the infinite geometric progress

\ [ \sum_ { n=1 } ^ { \infty } x^ { n-1 } = \frac { 1 } { 1-x } \mbox { for } | x | < 1.\ ] | (6) |

If \ ( x = 0, \ ) then \ ( \vert x\vert = \vert 0\vert < 1, \ ) which leads to

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\ [ \sum_ { n=1 } ^ { \infty } 0^ { n-1 } = \frac { 1 } { 1-0 } = 1.\ ] | (7) |

The infinite sum can be expanded as 00 + 01 + 02 + … = 1. As stated by Vaughan, if 00 is not defined, this summation is otiose. farther, if 00 ≠ 1, then the summation is assumed.

**Example 2.** This case arises from the infinite summation for antique, which can be written as

\ [ \sum_ { n=1 } ^ { \infty } \frac { x^ { n-1 } } { ( n-1 ) ! } = e^x \mbox {, for all } x.\ ] | (8) |

Everyone agrees that 0 ! = 1, indeed in the case where x = 0, the sum becomes

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\ [ \sum_ { n=1 } ^ { \infty } \frac { 0^ { n-1 } } { ( n-1 ) ! } = e^0 = 1.\ ] | (9) |

The sum can be expanded as

\ [ \frac { 0^0 } { 0 ! } + \frac { 0^1 } { 1 ! } + \frac { 0^2 } { 2 ! } + \cdots = \frac { 0^0 } { 1 } + 0 + 0 + \cdots = 0^0.\ ] | (10 |

The right-hand-side of the sum is e0 = 1, so 00 = 1.

**Example 3.** A third base exemplar given by Vaughan involves the cardinal number number of a determined of mappings. In set theory, exponentiation of a cardinal total is defined as follows :

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ab is the cardinal count of the set of mappings of a set up with b-complex vitamin members into a set with a members .

For case, 23 = 8 because there are eight ways to map the set { adam, y, omega } into the determined { a, boron }. In order to calculate 00, determine the number of mappings of the vacate bent into itself. There is precisely one such map, which is itself the set of the empty bent. “ thus, angstrom far as cardinal numbers are concerned, ” wrote Vaughan, “ 00 = 1. ”

When might a mathematician want 00 to be something that is not indeterminate ? If, for example, we are discussing the function degree fahrenheit ( x, y ) = xy, the origin is a discontinuity of the function. No matter what value may be assigned to 00, the routine xy can never be continuous at x = y = 0. Why not ? The limit of xy along the line x = 0 is 0, but the restrict along the line yttrium = 0 is 1, not 0. For consistency and utility, a “ natural ” choice would be to define 00 = 1 .