ABSOLUTELY AND CONDITIONALLY CONVERGENT SERIES

Egamnazarova  Jumagul Khudoyor qizi; Abduraximova Dilnoza

Authors

Egamnazarova Jumagul Khudoyor qizi Trainee Lecturer, Yangiyer Branch of Tashkent Institute of Chemical Technology Yangiyer, Uzbekistan
Abduraximova Dilnoza First-year Student, Food Technology Yangiyer Branch of Tashkent Institute of Chemical Technology

Keywords:

Absolutely convergent series,, conditionally convergent series,, infinite series,, Cauchy criterion,, convergence,, divergence,

Abstract

This article discusses the concepts of absolutely and conditionally convergent series, which play an important role in mathematical analysis. The relationship between a series and the series formed by the absolute values of its terms is examined. A fundamental theorem stating that absolute convergence implies convergence is presented and justified using the Cauchy criterion. The definitions of absolute and conditional convergence are provided along with illustrative examples. In particular, the alternating harmonic series is analyzed as a classical example of a conditionally convergent series, and its connection to the Maclaurin expansion of the logarithmic function is demonstrated. The results highlight the differences between absolute and conditional convergence and their significance in the study of infinite series.