  Self-similar sequence transformation is an original type of nonlinear
sequence transformations allowing for defining effective limits of asymptotic
sequences. The method of self-similar factor transformations is shown to be
regular. This method is applied for calculating the critical exponents of the
$O(N)$-symmetric $\varphi^4$ theory in three dimensions by summing asymptotic
$\varepsilon$ expansions. It is shown that this method is straightforward and
essentially simpler than other summation techniques involving complicated
numerical calculations, while enjoying comparable accuracy.
