The generalized Langevin equation is solved numerically by replacing the driving forces by stochastic, Gaussian distributed forces with a Gaussian time correlation. The calculated Brownian trajectories are compared with the corresponding classical mechanical and Monte Carlo trajectories and found to exhibit fractal properties with a dimension equal to two, for time intervals where the stochastic forces are uncorrelated. The classical mechanical trajectories have a lower fractal dimension, whereas the Markovian Monte Carlo trajectories show a fractal dimension equal to 2 even for high resolutions. The result indicates that the long-time memory plays a crucial role for the fractal dimensionality.