  The asymptotic iteration method (AIM) is applied to obtain highly accurate
eigenvalues of the radial Schroedinger equation with the singular potential
V(r)=r^2+\lambda/r^\alpha (\alpha,\lambda> 0) in arbitrary dimensions. Certain
fundamental conditions for the application of AIM, such as a suitable
asymptotic form for the wave function, and the termination condition for the
iteration process, are discussed. Several suggestions are introduced to improve
the rate of convergence and to stabilize the computation. AIM offers a simple,
accurate, and efficient method for the treatment of singular potentials such as
V(r) valid for all ranges of coupling \lambda.
