• Brima Conteh
• Christian Abu Sandy
• Momoh Kamara

The research explores the convergence properties of series in both real and complex number spaces, combining theoretical insights with computational verification. Traditional convergence tests—including comparison, ratio, root, and the Cauchy-Hadamard theorem—are re-examined and extended to accommodate complex series and functional spaces. MATLAB-based numerical simulations validate the effectiveness of these classical methods, showing rapid convergence of geometric and exponential series within their expected limits and divergence at boundary conditions, thereby confirming the reliability of the criteria. Notably, the exponential series was demonstrated to converge uniformly across the entire complex plane, aligning with the characteristics of entire functions, while the radius of convergence effectively predicts the behaviour of power series over various domains. Overall, the findings affirm that classical convergence criteria remain applicable and robust in complex analysis, emphasizing the critical role of the radius of convergence. The study also underscores the practical significance of these criteria in fields such as physics and engineering, particularly for boundary value problems.

Convergence, Real Series, Complex Series

"Analysis of Convergence of Series in Real and Complex Series", International Journal of Emerging Technologies and Innovative Research (www.jetir.org), ISSN:2349-5162, Vol.12, Issue 10, page no.b473-b477, October-2025, Available :http://www.jetir.org/papers/JETIR2510165.pdf

"Analysis of Convergence of Series in Real and Complex Series", International Journal of Emerging Technologies and Innovative Research (www.jetir.org | UGC and issn Approved), ISSN:2349-5162, Vol.12, Issue 10, page no. ppb473-b477, October-2025, Available at : http://www.jetir.org/papers/JETIR2510165.pdf