the following link can answer your general questions about improper integrals ...

Pauls Online Notes : Calculus II - Improper Integrals

for your specific example, start by noting the following inequality ...

for the larger integral, let

substitute ...

using integration by parts, it's a fairly simple process to show the convergence of this integral.

since the improper integral in question is less than this convergent improper integral, by direct comparison they both converge.