We study the equations of motion of the massive and massless particles in the Schwarzschild geometry of general relativity by using the Laplace-Adomian Decomposition Method, which proved to be extremely successful in obtaining series solutions to a wide range of strongly nonlinear differential and integral equations. After introducing a general formalism for the derivation of the equations of motion in arbitrary spherically symmetric static geometries and of the general mathematical formalism of the Laplace-Adomian Decomposition Method, we obtain the series solution of the geodesics equation in the Schwarzschild geometry. The truncated series solution, containing only five terms, can reproduce the exact numerical solution with a high precision. In the first order of approximation we reobtain the standard expression for the perihelion precession. We study in detail the bending angle of light by compact objects in several orders of approximation. The extension of this approach to more general geometries than the Schwarzschild one is also briefly discussed.