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Gregory's Series : Taylor Expansion, Interval of Convergence and Calculation of 'pi'

In 1671, James Gregory, a Scottish mathematician, developed the following series for tan^-1 x
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1. Verify that Gregory's series is correct by using a Taylor Series expansion or methods of power series.

2. Find the interval of convergence of Gregory's series.

3. Using Gregory's series, find a series whose sum is pi/4 by assigning a value of 1 to x.

4. Abraham Sharp in 1699, and DeLangy in 1719 found values for pi correct to 71 decimal places and 112 decimal places, respectively, using Gregory's series by substituting x = sqrt(1/3). Find the series that they used.

The information used in this set of problems comes from "An Introduction to the History of Mathematics", Fourth Edition by Howard Eves, page 99.

Please see the attached file for the fully formatted problems.


Solution Summary

Gregory's Series is examined using a Taylor Expansion, Interval of Convergence and Calculation of 'pi'. The solution is detailed and well presented.