Chapter 9, Problem 49RE

To determine

Whether the given integral is convergent or divergent.

Answer to Problem 49RE

The given integral is divergent.

Explanation of Solution

Given information :

  ${\displaystyle \underset{1}{\overset{\infty }{\int }}\frac{\ln z}{z}}dz$

Formula used :

Comparison test

Calculation :

Let, $f\left(z\right)=\frac{\ln z}{z}$ and $g\left(z\right)=\frac{1}{z}$

It is obvious that $f\left(z\right)>g\left(z\right)\forall \text{z}\in \left(1,\infty \right)$

Here, ${\displaystyle \underset{1}{\overset{\infty }{\int }}g\left(z\right)}dz$ is divergent

So, $f\left(z\right)$ is also divergent.