Very important concepts in both pure mathematics and engineering are infinite series convergence. The study examines the fundamental concepts of the convergence criteria, including the absolute and conditional convergence, and the theories underlying convergence criteria. This paper discusses such common tests as the Ratio test, the Root test and the Comparison test to demonstrate when an infinite series is sure to converge safely. These convergence virtues can be traced in the engineering fields such as signal processing, control systems and numerical approaches where infinite series solution models complex events. The usefulness is highlighted in the study through its connection of mathematical theory and applications to the engineering field and illustration of how convergence ensures that systems remain stable and computations made correct.