Chapter 1.6, Problem 16P

In testing ${\displaystyle \sum 1/n^2}$ for convergence, a student evaluates ${\displaystyle {\int }_0^{\infty }{n}^{-2}dn=-n^{-1}}|0^{\infty }=0+\infty =\infty$ and concludes (erroneously) that the series diverges. What is wrong? Hint: Consider the area under the curve in a diagram such as Figure 6.1 or 6.2. This example shows the danger of using a lower limit in the integral test.