  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.
