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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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