Local Precision and Global Harmony: A Comparative Literature Review (LR) Framework for Taylor and Fourier Series in Engineering Modeling
Keywords:
Arithmetic series, geometric series, power series, Fourier series, Taylor series, convergence, periodic functions, series expansion, mathematical analysis, signal processing.Abstract
Mathematical series are foundational tools in both theoretical and applied sciences, yet a coherent, comparative understanding of their distinct behaviors especially in engineering contexts remains underexplored. This paper addresses a critical research gap by systematically investigating how and why different series types (arithmetic, geometric, power, Taylor, and Fourier) exhibit fundamentally divergent convergence properties, representational capabilities, and domain-specific efficacies when modeling real-world phenomena. The central research question is: Under what conditions should a given series type be preferred for engineering analysis, particularly when dealing with periodicity, discontinuities, or local versus global behavior?. The novelty of this work lies in its integrative framework that contrasts local approximations (Taylor/power series) with global, periodic representations (Fourier series), explicitly linking mathematical theory to engineering practice. This research study demonstrates that while Taylor series excel in high-precision local modeling near analytic points, they fail to represent discontinuous or periodic systems common in mechanical vibrations, signal processing, and thermal dynamics. In contrast, Fourier series robustly capture such behaviors through harmonic decomposition, despite exhibiting the Gibbs phenomenon near discontinuities. This research contributes directly to the engineering domain by providing a decision-oriented taxonomy that guides practitioners in selecting the optimal series expansion for problems in vibration analysis, heat transfer, fault diagnostics, acoustics, and control systems. By clarifying convergence limitations, operational rules (e.g., term-wise differentiation and integration), and the physical interpretability of harmonic components, this review enhances both pedagogical clarity and computational reliability in engineering modeling. Ultimately, the work bridges abstract mathematical theory and applied engineering design, empowering more informed, effective use of series-based methods in modern technological challenges.
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