Artificial intelligence

Abstract: In the paper, two forms of limiting comparison tests for estimation of convergence of the number series and improper integrals are proposed. It is shown that the Riemann zeta-function allows to formulate the new necessary test of convergence of the series to get such well-known tests as Cauchy’s Root Test and d’Alembert’s Ratio Test. The new test with a parameter allows to use l’Hopital’s rule more effectively.

Keywords: series, convergence, zeta-function, comparison tests, l’Hopital’s rule, limit, integral test

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