Machin-like formula
Formulas for pi / From Wikipedia, the free encyclopedia
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In mathematics, Machin-like formulas are a popular technique for computing π (the ratio of the circumference to the diameter of a circle) to a large number of digits. They are generalizations of John Machin's formula from 1706:
which he used to compute π to 100 decimal places.[1][2]
Machin-like formulas have the form
(1) |
where is a positive integer, are signed non-zero integers, and and are positive integers such that .
These formulas are used in conjunction with Gregory's series, the Taylor series expansion for arctangent:
(2) |