Study on the Convergence and Divergence of Positive Series in Split-Complex Analysis

Year 2025, Issue: 53, 24 - 35, 31.12.2025

Zhishang Huang , Liudi Peng , Yongyi Gu

https://doi.org/10.53570/jnt.1763538

Abstract

Split-complex numbers $\mathbb{R}^{1,1}$ are an extension of the real numbers $\mathbb{R}$, forming a commutative ring generated by two real numbers and featuring two zero divisors. By utilizing the properties of split-complex numbers, this paper mainly presents criteria for the convergence of series with positive terms in split-complex analysis, including proofs of the comparison criterion, the D'Alembert criterion, the Cauchy criterion, the Cauchy-Hadamard criterion, and the Raabe criterion on $\mathbb{R}^{1,1+} \setminus \{0\}$. The results demonstrate that the general criteria for series with positive terms in real analysis are still applicable in split-complex analysis. The results obtained further refine the theoretical framework of split-complex numbers.

Keywords

Split-complex numbers , comparative criterion on $\mathbb{R}^{-1,1+}\setminus\{0\}$ , D'Alembert's criterion on $\mathbb{R}^{1,1+}\setminus\{0\}$ , Cauchy criterion on $\mathbb{R}^{1,1+}\setminus\{0\}$ , Raabe criterion on $\mathbb{R}^{1,1+}\setminus\{0\}$

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