How to see this improper integral diverges? $$\int^\infty_1\frac{1}{x^{1+1/x}}dx$$
I'm preparing for exams. I would also like to know what are some commonly used methods to show an improper integral diverges?

$$
\begin{align}
\int_n^\infty\frac1{x^{1+1/x}}\,\mathrm{d}x
&\ge\int_n^\infty\frac1{x^{1+1/n}}\,\mathrm{d}x\\
&=nn^{-1/n}
\end{align}
$$
and
$$
\begin{align}
\lim_{n\to\infty}nn^{-1/n}
&=\lim_{n\to\infty}n\lim_{n\to\infty}n^{-1/n}\\
&=\infty\cdot1
\end{align}
$$

Another approach is to notice that $x^{1/x}\le e^{1/e}$.
