Chapter 6.2, Problem 115E

In the following exercises, using a substitution if indicated, express each series in terms of elementary functions and find the radius of convergence of the sum.

115. [T] Recall that ${\tan }^{-1}\left(\frac{1}{\sqrt{3}}\right)=\frac{\pi }{6}$ . Assuming an exact value of $\left(\frac{1}{\sqrt{3}}\right)$ , estimate $\frac{\pi }{6}$ by evaluating partial sums S_N $\left(\frac{1}{\sqrt{3}}\right)$ of the power series expansion ${\tan }^{-1}(x)={\displaystyle \underset{k=0}{\overset{\infty }{\sum }}{(-1)}^k\frac{x^{2k+1}}{2k+1}\text{at}\frac{1}{\sqrt{3}}}$ . What is the smallest number N such that $6S_N\left(\frac{1}{\sqrt{3}}\right)$ approximates $\pi$ accurately so within 0,001? How many term are needed for accuracy so within 0.00001?