Cauchy's Criterion for Non-negative Hyperbolic Series

Hyperbolic number plane constitutes a significant area of algebraic research, with their properties finding wide-ranging applications across various fields. Meanwhile, Cauchy's criterion, as an effective tool for determining series convergence, has also been extensively utilized in the study of hyperbolic numbers within mathematical analysis and practical problem-solving, offering valuable insights and practical applications. The paper conducted an in-depth exploration of the non-negative hyperbolic plane, applying the Comparison test and further investigating the Cauchy's criterion on the non-negative hyperbolic plane. Through the application of the comparison test, the paper have arrived at an important conclusion: when considering two numbers, and , in the non-negative hyperbolic complex plane, and , the convergence of implies the convergence of , and divergence of implies the divergence of . Furthermore, through a detailed analysis of Cauchy’s criterion, the paper also discovered a key finding: when a certain number r in the non-negative hyperbolic plane satisfies , if , the corresponding infinite series converges, whereas if , the series diverges. The paper not only pushes the frontiers of mathematical theory but also unlocks fresh avenues for interdisciplinary exploration. By delving into uncharted territories, it expands our understanding of mathematical principles and their applicability across diverse fields. Moreover, it fortifies mathematics' central role in addressing real-world challenges, providing essential tools and insights. As a cornerstone for future research, this work paves the way for innovative mathematical investigations and their practical applications, fostering a deeper symbiosis between theory and the real world.