Chapter 8.6, Problem 33E

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.

33. $\frac{7{\pi }^4}{720}={\displaystyle \underset{k=1}{\overset{\infty }{\sum }}\frac{{(-1)}^{k+1}}{k^4}}$