Using series for determining the convergence of improper integral

Question

We apply the integral test for determining the convergence of series . For example integral test is useful for $\sum_{n=1}^\infty \frac{1}{n}$ or $\sum_{n=1}^\infty \frac{\ln n}{n}$ . Because the integral test is biconditional , I thought about the other way . I mean determining the convergence of the improper integral with the help of series . Is it practical ? What are the examples for it ?

Answer 1

Sure. In some cases, it is practical. Consider, for instance, the integral $\displaystyle\int_2^\infty\frac1{x(x-1)}\,\mathrm dx$. Then\begin{align}\int_2^\infty\frac1{x(x-1)}\,\mathrm dx\text{ converges}&\iff\sum_{n=2}^\infty\frac1{n(n-1)}\text{ converges}\\&\iff\sum_{n=2}^\infty\left(\frac1n-\frac1{n+1}\right)\text{ converges,}\end{align}and it is clear that the last statment holds (it's a telescoping series).