Averaging an ODE of the form \(\dot {x}= \varepsilon f(t, x)\) , the averaged equation will, in many cases, contain equilibria (stationary solutions). Under rather general conditions, these stationary solutions will have in an \(\varepsilon \) -neighbourhood a periodic solution of the original equation. Examples are a generalised Van der Pol-equation and the Duffing-equation with small forcing. The Poincaré-Lindstedt method is introduced to obtain a convergent series for these periodic solutions.