We consider a summation technique which reduces summation of a series to the solution of some linear functional equations. Partial sums of a series satisfy an obvious difference equation. This equation is transformed to the functional equation on the interval [0,1] for the continuous argument. Then this equation is either solved explicitly (to within an arbitrary constant) or an asymptotic expansion of the solution is computed at the origin. The sum of the original series is determined uniquely as a constant needed for the matching of the asymptotic series with partial sums of the original series. The notion of a limit is not involved in this computational technique, which allows summation of divergent series as well.

summation of series, functional equations, divergent series

Mathematical problems and theory of numerical methods

Varin Victor Petrovich,  ,  orcid.org/0000-0001-7324-1823,  KIAM RAS