Chapter 9.6, Problem 47ES

The purpose of this exercise is to show that the error bound in part (b) of Theorem 9.6.2 can be overly conservative in certain cases.

(a) Use a CAS to confirm that $\frac{\pi }{4}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots$

(b) Use the CAS to show that $\left|\left(\pi /4\right)-s_{25}\right|<{10}^{-2}.$

(c) According to the error bound in part (b) of Theorem 9.6.2, what value of n is required to ensure that $\left|\left(\pi /4\right)-s_n\right|<{10}^{-2}?$