11-00-GregorySeries.tex

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\begin{document}
The {\em Gregory series} is an alternating sum whose value is a quarter that of $\pi$: $$\frac{\pi}{4} = \sum_{i = 0}^\infty (-1)^i \frac{1}{2i + 1} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \ldots$$
(The approximate decimal value of this expression is 0.7853981633974483...)

More generally, a Gregory series for a given $n$ is $$\sum_{i = 0}^\infty (-1)^i \frac{n^{2i + 1}}{2i + 1}.$$

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