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

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In 1671, James Gregory, a Scottish mathematician, developed the following series for tan^-1 x
{See attachment}

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

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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 = 

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

Proof. Since , we ...

Solution provided by:
Changping Wang, MA
Education
  • BSc , Wuhan Univ. China
  • MA, Shandong Univ.
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  • "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
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