The improved understanding of dynamical systems coming from chaos theory helped shed light on what was termed the small denominator problem or small divisor problem. In the 19th century Poincaré observed (as perhaps had earlier mathematicians) that sometimes 2nd and higher order terms in the perturbative series have "small denominators": That is, they have the general form ψ ψ n V ϕ ϕ m ( ω ω n − − ω ω m ) where ψ ψ n , V , and ϕ ϕ m are some complicated expressions pertinent to the problem to be solved, and ω ω n and ω ω m are real numbers; very often they are the energy of normal modes. The small divisor problem arises when the difference ω ω n − − ω ω m is small, causing the perturbative correction to "blow up", becoming as large or maybe larger than the zeroth order term. This situation signals a breakdown of perturbation theory: It stops working at this point, and cannot be expanded or summed any further. In formal terms, the perturbative series is an asymptotic series : A useful approximation for a few terms, but at some point becomes less accurate if even more terms are added. The breakthrough from chaos theory was an explanation of why this happened: The small divisors occur whenever perturbation theory is applied to a chaotic system. The one signals the presence of the other.