# 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.

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

In 1671, James Gregory, a Scottish mathematician, developed the following series for tan-1 x:

tan-1x = ïƒ¥

1. Verify that Gregory's series is correct by using a Taylor Series expansion or methods of power series.

Proof. Since , we ...

#### 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.