In this article, a residual power series technique for the power series solution of systems of initial value problems is introduced. The new approach provides the solution in the form of a rapidly convergent series with easily computable components using symbolic computation software. The proposed technique obtains Taylor expansion of the solution of a system and reproduces the exact solution when the solution is polynomial. Numerical examples are included to demonstrate the efficiency, accuracy, and applicability of the presented technique. The results reveal that the technique is very effective, straightforward, and simple.