Finding interval of convergence of gregorys series

Question

Assignment:

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 = Σ^∞₀  (-1)ⁿx^2n-1 / 2n+1

Q1. Verify that Gregory’s series is correct by using a Taylor Series expansion or methods of power series.

Q2. Find the interval of convergence of Gregory’s series.

Q3. Using Gregory’s series, find a series whose sum is π/4 by assigning a value of 1 to x.

Q4. Abraham Sharp in 1699, and DeLangy in 1719 found values for π correct to 71 decimal places and 112 decimal places, respectively, using Gregory’s series by substituting x = √(1/3).  Find the series that they used.  Hint: note that tan^-1√(1/3).  = (π/6)

Provide complete and step by step solution for the question and show calculations and use formulas.