Self-similar continued root approximants

A novel method of summing asymptotic series is advanced. Such series repeatedly arise when employing perturbation theory in powers of a small parameter for complicated problems of condensed matter physics, statistical physics, and various applied problems. The method is based on the self-similar approximation theory involving self-similar root approximants. The constructed self-similar continued roots extrapolate asymptotic series to finite values of the expansion parameter. The self-similar continued roots contain, as a particular case, continued fractions and Pad\'{e} approximants. A theorem on the convergence of the self-similar continued roots is proved. The method is illustrated by several examples from condensed-matter physics.