ERROR ANALYSIS IN NUMERICAL APPROXIMATION USING TAYLOR’S THEOREM

Numerical methods are widely employed to approximate solutions to mathematical problems that are analytically intractable. However, these approximations inevitably introduce errors arising from truncation, rounding, and error propagation, which, if left unexamined, may compromise the reliability of computed results. This project investigates the nature of such errors within the framework of numerical approximation. Beginning with Taylor’s theorem and its remainder, the study derives fundamental numerical methods for differentiation, integration, and the solution of ordinary differential equations. The associated error terms and bounds are rigorously established, thereby validating the accuracy and order of each method. The analysis demonstrates how truncation errors can be quantified, how error bounds provide guarantees on approximation quality, and how convergence is ensured in practical computations. By applying the Taylor series method, the project highlights the behavior of errors in numerical solutions and illustrates strategies for controlling them. The findings confirm that numerical methods, when rigorously analyzed through Taylor expansions and error bounds, provide reliable approximations with predictable accuracy, thereby addressing the challenges identified in the study.