Analysis of algebraic and geometric ideas in infinite series

Abstract As an important concept in mathematics, the convergence judgment of series has always been one of the research hotspots. As the reference series to judge the convergence and divergence of series, the proof of convergence of P-series and harmonic series has always been a research hotspot. This paper discusses the embodiment of algebra and geometry in the proof of convergence and divergence of P-series and harmonic series. Through the mutual transformation of geometry and algebra, the convergence and divergence of infinite series are analyzed, and the convergence problem of series is transformed into an intuitive geometric image for analysis. By introducing geometric intuition, it can not only simplify the solution process of the problem, but also help students and researchers better understand the essential characteristics of series, so as to provide new perspectives and ideas for the study of series convergence.