Abstract
We describe a simple analytical method for effective summation of series, including divergent series. The method is based on self-similar approximation theory resulting in self-similar root approximants. The method is shown to be general and applicable to different problems, as is illustrated by a number of examples. The accuracy of the method is not worse, and in many cases better, than that of Padé approximants, when the latter can be defined.
Highlights
In numerous cases of applied mathematics and mathematical physics the solutions to problems can only be represented as series derived by means of some kind of perturbation theory or iterative procedure
We have described a simple and general method for interpolating functions between their small-variable and large-variable asymptotic expansions
The method is based on the construction of self-similar root approximants enjoying the general form fk∗ (x) = f0 (x)
Summary
In numerous cases of applied mathematics and mathematical physics the solutions to problems can only be represented as series derived by means of some kind of perturbation theory or iterative procedure. The standard way of treating such asymptotic series, for the purpose of their extrapolation to the finite values of the variable, is by invoking the Padé approximants [1] The latter, exhibit several deficiencies limiting their applicability, as is discussed in References [1,2], for instance, such a notorious deficiency as the appearance of spurious poles. The method enjoys the following advantages: (i) It is unambiguously defined for each given series of order k; (ii) It allows for the treatment of large-variable behavior of any type, whether with integer, rational, or irrational powers; (iii) Being more general, it is not less accurate than the method of the Padé approximants, when the latter exist, in many cases, being more accurate. In the examples below, we do not consider very large series, showing that even several terms allow us to derive quite accurate approximations
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
