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    Ratio Test

    Alternate names

    d'Alembert's ratio test | Cauchy ratio test

    Theorem

    Let a_n be a sequence of real numbers: 1.If lim_(n->∞) abs(a_(n + 1)/a_n)<1, then the series sum_(n=1)^∞ a_n is absolutely convergent and therefore convergent. 2.If lim_(n->∞) abs(a_(n + 1)/a_n)>1, then the series sum_(n=1)^∞ a_n is divergent. 3.If lim_(n->∞) abs(a_(n + 1)/a_n) = 1, then the test is inconclusive and cannot determine whether the series sum_(n=1)^∞ a_n is convergent or divergent.

    Details

    limit of a sequence | convergent series | absolutely convergent series | divergent series

    comparison test | limit comparison test | integral test | root test | alternating series test

    Associated people

    Jean Le Rond d'Alembert | Augustin-Louis Cauchy

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