  The Asymptotic Iteration Method (AIM) is a technique for solving analytically
and approximately the linear second-order differential equation, especially the
eigenvalue problems that frequently appear in theoretical and mathematical
physics. The analysis and mathematical justifications of the success and
failure of the asymptotic iteration method are detailed in this work. A theorem
explaining why the asymptotic iteration method works for the eigenvalue problem
is presented. As a byproduct, a new procedure to generate unlimited classes of
exactly solvable differential equations is also introduced.
