[en] A method for summation of alternating asymptotic series which is based on a modified Borel transformation with a Borel transform B(z) which is an entire function of z is proposed. The method is designed to produce results when B(z) exhibits a power-law asymptotic for Z large and positive. The method is illustrated on the examples of the groundstate energy level of the quantum-mechanical anharmonic oscillator gx4, the critical exponents and the Gell-Mann-Low function in g(PHI vector2)2-field theory