Chapter 9.4, Problem 38ES

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 37-40.

In each part, find upper and lower bounds on the error that results if the sum of the series is approximated by the 10th partial sum.

$\begin{array}{ccc}\left(\text{a}\right){\displaystyle \underset{k=1}{\overset{\infty }{\sum }}\frac{1}{{\left(2k+1\right)}^2}}& \left(\text{b}\right){\displaystyle \underset{k=1}{\overset{\infty }{\sum }}\frac{1}{k^2+1}}& \left(\text{c}\right){\displaystyle \underset{k=1}{\overset{\infty }{\sum }}\frac{k}{e^k}}\end{array}$