PlanetMath: Gregory series

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Gregory series

(Definition)

The Gregory series is an alternating sum whose value is a quarter that of $\pi$ :

$\displaystyle \frac{\pi}{4} = \sum_{i = 0}^\infty (-1)^i \frac{1}{2i + 1} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \ldots$

(The approximate decimal value of this expression is 0.7853981633974483...)

More generally, a Gregory series for a given is

$\displaystyle \sum_{i = 0}^\infty (-1)^i \frac{n^{2i + 1}}{2i + 1}.$

The Gregory series is named after the Scottish astronomer and astrologer James Gregory.

"Gregory series" is owned by PrimeFan.

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Cross-references: expression, alternating sum
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This is version 1 of Gregory series, born on 2007-10-19.
Object id is 10005, canonical name is GregorySeries.
Accessed 383 times total.

Classification:

AMS MSC:	11-00 (Number theory :: General reference works )
	51-00 (Geometry :: General reference works )
	01A16 (History and biography :: History of mathematics and mathematicians :: Egyptian)
	01A20 (History and biography :: History of mathematics and mathematicians :: Greek, Roman)
	01A25 (History and biography :: History of mathematics and mathematicians :: China)
	01A32 (History and biography :: History of mathematics and mathematicians :: India)
	01A40 (History and biography :: History of mathematics and mathematicians :: 15th and 16th centuries, Renaissance)

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