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'''Gregory's series''', also known as the '''Madhava-Gregory series''' or '''Leibniz's series''', is a mathematical [[Series (mathematics)|series]] that was discovered by the Indian mathematician [[Madhava of Sangamagrama]] (1350- 1410) {{harv|Gupta|1973}}. Later the series was rediscovered and published in 1668 by [[James Gregory (mathematician)|James Gregory]]. In fact, it was rediscovered by both Gregory and Leibniz and obtained by plugging x=1 into the [[Leibniz series]].<ref name=wolfram>{{cite web|title=Gregory Series|url=http://mathworld.wolfram.com/GregorySeries.html|publisher=Wolfram Math World|accessdate=26 July 2012}}</ref> 

==Definition==
Gregory's series is defined as below:

: <math>\int_0^x \, \frac{du}{1+u^2} =  \arctan x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots </math>

Compare with the [[Sine#Series definition|series for sine]], which is similar but has factorials in the denominator.

==History==
In 1668 Gregory published two works. One of them, ''Geometriae pars universalis'' (The Universal Part of Geometry), was published in [[Padua]]; the other, ''Exercitationes geometrica'' (Geometrical Exercises), at [[London]]. He divided mathematics into "general" and "special" groups of theorems. He discovered the [[Taylor series]] more than forty years before [[Brook Taylor]] published it. The series for arctan ''x'' bears his name.

==See also==
* [[List of mathematical series]]

==References==
{{Reflist}}
* Carl B. Boyer, A history of mathematics, 2nd edition, by John Wiley & Sons, Inc., page 386, 1991
* {{cite journal|first=RC|last=Gupta|title=The Madhava&ndash;Gregory series|journal=Mathematical Education|volume=7|year=1973|pages=67&ndash;70|ref=harv}}

[[Category:Mathematical series]]