Chapter 8.6, Problem 31E

Remainders in alternating series Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than 10⁻⁴ in magnitude. Although you do not need it, the exact value of the series is given in each case.

31. $\frac{\pi }{4}={\displaystyle \underset{k=0}{\overset{\infty }{\sum }}\frac{{(-1)}^k}{2k+1}}$