Gregory series

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

$$\frac{\pi }{4}=\sum _{i=0}^{\mathrm{\infty }}{(-1)}^i\frac{1}{2i+1}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\mathrm{\dots }$$

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

More generally, a Gregory series for a given $n$ is

$$\sum _{i=0}^{\mathrm{\infty }}{(-1)}^i\frac{n^{2i+1}}{2i+1}.$$

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

Title	Gregory series
Canonical name	GregorySeries
Date of creation	2013-03-22 17:35:33
Last modified on	2013-03-22 17:35:33
Owner	PrimeFan (13766)
Last modified by	PrimeFan (13766)
Numerical id	4
Author	PrimeFan (13766)
Entry type	Definition
Classification	msc 11-00
Classification	msc 51-00
Classification	msc 01A16
Classification	msc 01A20
Classification	msc 01A25
Classification	msc 01A32
Classification	msc 01A40
Related topic	TaylorSeriesOfArcusTangent