Consider this integral:

\[ \int_2^\infty \frac{1}{x(\ln x)^p}\textrm{d} x . \]

1. Use R to create a single plot with many lines, each graphing the integrand for a different value of \(p\). Consider \(p\) equal to -2, -1.5, -1, 0, 1, and 5, and make the \(x\)-axis of your plot run from 2 to 15.
2. Show that \(\lim_{x\to\infty}\frac{1}{x(\ln x)^p}=0\) for all values of \(-\infty<p<\infty\).
3. For what values of \(p\) does the integral converge? When it does converge, what is its value?
4. Consult the picture you created in part (a), and write a few sentences explaining conceptually why the integral converges for some values of \(p\) but not others.

Hint

When taking the limit or evaluating the integral, can you use the same technique for all values of \(p\), or do you need a different technique depending on what \(p\) is?