Theorem on convergence of a formal solution of ODE

Here is presented a proof of the theorem, which was formulated in more general form by A.D. Bruno in the paper in “Russian Mathematical Surveys” (2004). Proof is realized for formal solution of non-linear ordinary differential equation in general form, which is the series with complex power exponents and constant coefficients. Here we consider the case of the one complex generatrix in the set of power exponents of the series. Theorem is proved by the majorant method and new approaches using methods of plain and three-dimensional Power Geometry.