Many active sound or vibration control applications need adaptation to achieve their goals. Nowadays there are two algorithms that are generally used to perform adaptation in such applications: Recursive Least Squares algorithm (RLS) and Least Mean Squares algorithm (LMS). The former, however computationally more expensive, is better known from theoretical point of view. The latter, very simple to implement, still lacks complete theoretical analysis. The paper presents recent author discoveries in convergence analysis of LMS algorithm. These results allowed to formulate new necessary condition on convergence of LMS algorithm. It must be emphasized that this condition was obtained without small step assumption that is crucial in many theoretical LMS analyses presented so far. Moreover, the condition proved to be relatively easy to use in practice. Based on the new convergence condition the algorithm was derived allowing for fast adaptation with the guarantee of convergence. The paper presents theoretical basis for the algorithm as well as the results of simulation experiments showing its excellent performance.