Rational summation of $p$-adic series

Problem of the rational summation for a wide class of $p$-adic convergent series is considered. Here, rational summation means a method to obtain a rational sum of power series for a rational value of its variable. Formula suitable for this summation is derived. Conditions for rational summability are obtained. Rational summation is possible only for special forms of the series. It is shown that the inverse problem of rational summation is always solvable. This is illustrated by some characteristic examples. Possible rational (adelic) summation of divergent perturbative expansions in string theory, and quantum field theory, is discussed.