CONVERGENCE TESTS FOR SERIES: D'ALEMBERT'S TEST AND CAUCHY'S TESTS

This paper discusses two main convergence tests for series: D'Alembert's test (or D'Alembert's criterion) and Cauchy's test. D'Alembert's test is based on the analysis of the ratio of adjacent terms of a series and allows one to establish the convergence or divergence of a series depending on the behavior of this ratio as the index tends to infinity. In turn, Cauchy's test, also known as the Cauchy criterion, uses the concept of the limit of partial sums and provides a condition for the convergence of a series through the analysis of its partial sums and their behavior. Both of these tests are powerful tools for studying the convergence of numerical series and are widely used in mathematical analysis. Their formulations, application examples, and features of use in various problems are considered.