Use Direct Comparison test to show Divergence

Question

4.Use Direct Comparison test to show that :

$$\int_1^\infty\cfrac{1+e^{-x}}{x}dx \qquad \qquad (a)$$ Diverges

Is equal to :

\begin{align} & =\int_1^\infty \cfrac{1+\cfrac{1}{e^x}}{x}dx \\ & = \int_1^\infty \cfrac{\cfrac{e^x+1}{e^x}}{x}dx\\ & = \int_1^\infty \cfrac{e^x+1}{xe^x}dx\\ \end{align}

I also done $xe^x\geqslant e^x+1$

In this step i've thinked of doing a substitution, but drives me nowhere, for what function is convenient to compare ?

Answer 1

Hint: $$\frac{1+e^{-x}}{x}\ge\frac1x\quad\forall x\ge1$$

Answer 2

You can use the limit comparison, comparing with \begin{equation*} \frac{1}{x}. \end{equation*}