COMPARATIVE ANALYSIS OF THE D'ALEMBERT, CAUCHY, RAABE, AND GAUSS TESTS FOR SERIES WITH POSITIVE TERMS

This article presents a comparative mathematical analysis of the d’Alembert ratio test, Cauchy root test, Raabe test, and Gauss test for determining the convergence or divergence of infinite series with positive terms. The study investigates the theoretical foundations, mathematical conditions, analytical capabilities, and limitations of these classical convergence criteria. Particular attention is devoted to their effectiveness in examining series with exponential, polynomial, logarithmic, and asymptotically defined terms. The research employs theoretical analysis, asymptotic expansion, comparative mathematical reasoning, and representative examples to identify the conditions under which each test provides a definitive conclusion. The results demonstrate that the d’Alembert and Cauchy tests are particularly effective for series exhibiting exponential decay or growth, whereas the Raabe and Gauss tests provide more refined criteria for series whose consecutive-term ratios approach unity. The analysis also establishes that the Gauss test can distinguish certain borderline cases in which the Raabe test remains inconclusive.