Chapter 9.4, Problem 37ES

Exercise 36 will show how a partial sum can be used to obtain upper and lower bounds on the sum of a series when the hypotheses of the integral test are satisfied. This result will be needed in Exercises 37- (a)

(a) It was stated in Exercise 35 that ${\displaystyle \underset{k=1}{\overset{\infty }{\sum }}\frac{1}{k^2}=\frac{{\pi }^2}{6}}$

Show that if $s_n$ is the $n\text{th}$ partial sum of this series, then $s_n+\frac{1}{n+1}<\frac{{\pi }^2}{6}<s_n+\frac{1}{n}$

(b) Calculate $s_3$ exactly, and then use the result in part (a) to show that $\frac{29}{18}<\frac{{\pi }^2}{6}<\frac{61}{36}$

(c) Use a calculating utility to confirm that the inequalities in part (b) are correct.

(d) Find upper and lower bounds on the error that results if the sum of the series is approximated by the 10th partial sum.