If the power series in ε ε converges with a nonzero radius of convergence, the perturbation problem is called a regular perturbation problem. In regular perturbation problems, the asymptotic solution smoothly approaches the exact solution. However, the perturbation series can also diverge, and the truncated series can still be a good approximation to the true solution if it is truncated at a point at which its elements are minimum. This is called an asymptotic series. If the perturbation series is divergent or not a power series (for example, if the asymptotic expansion must include non-integer powers ε ε ( 1 / 2 ) or negative powers ε ε − − 2 ) then the perturbation problem is called a singular perturbation problem. Many special techniques in perturbation theory have been developed to analyze singular perturbation problems.