Self-similar bootstrap of divergent series

A method is developed for calculating effective sums of divergent series. This approach is a variant of the self-similar approximation theory. The interesting component here is using an algebraic transformation with a power providing maximal stability of the self-similar renormalization procedure. The latter is to be repeated as many times as necessary in order to convert into closed self-similar expressions all sums from the series considered. This multiple and complete renormalization is called a self-similar bootstrap. The method is illustrated by several examples from statistical physics.