Particle swarm optimization (PSO) is the most important and popular algorithm to solving the engineering optimization problem due to its simple updating formulas and excellent searching capacity. This algorithm is one of evolutionary computations and is also a population-based algorithm. Traditionally, to demonstrate the convergence analysis of the PSO algorithm or its related variations, simulation results in a numerical presentation are often given. This way may be unclear or unsuitable for some particular cases. Hence, this paper will adopt the illustration styles instead of numeric simulation results to more clearly clarify the convergence behavior of the algorithm. In addition, it is well known that three parameters used in the algorithm, i.e., the inertia weight w, position constants c1 and c2, sufficiently dominate the whole searching performance. The influence of these parameter settings on the algorithm convergence will be considered and examined via a simple two-dimensional function optimization problem. All simulation results are displayed using a series of illustrations with respect to various iteration numbers. Finally, some simple rules on how to suitably assign these parameters are also suggested