Problem

Part I Multiple Choice

5. Consider the improper integral $I_p=\int_0^\infty \dfrac{1}{x^p}\,\mathrm{d}x$, where $p$ is a positive number.

(A) $I_p$ is convergent if $p>1$ and divergent if $0<p\le 1$.

(B) $I_p$ is convergent if $0<p<1$ and divergent if $p\ge 1$.

(C) $I_p$ is convergent for all $p>0$.

(D) $I_p$ is divergent for all $p>0$.