The proof of an equivalence of D’alembert’s and Cauchy’s tests in the theory of numerical series

摘要

In the paper it is shown that D’alembert’s test and Cauchy’s radical test are not independent one from another. Itudis proposed a transition scheme from one test to another and vice versa. The equivalence of the tests is provedudfor series with the monotonically decreasing terms. This fact is used to formulate a new test of theudconvergence for series with positive terms. The test is equivalent to Dalembert and Cauchy’s radical tests, butudit has some advantages. It can be applied to any series within Cauchy-D’alembert’s theory.