Chapter 6.2, Problem 114E

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.

114. [T] Given the power series expansion ${\tan }^{-1}(x)={\displaystyle \underset{k=0}{\overset{\infty }{\sum }}{(-1)}^k\frac{x^{2k+1}}{2k+1}}$ , use the alternating series test to determine how many terms N of the sum evaluated at

x = 1 are needed to approximate tan-1(1) = $\frac{\pi }{4}$ accurate to within 1/1000. Evaluate the corresponding partial stun ${\displaystyle \underset{k=0}{\overset{\infty }{\sum }}{(-1)}^k\frac{x^{2k+1}}{2k+1}}$ .