Ratio test

In mathematics, the ratio test is a criterion for convergence or divergence of a series whose terms are real or complex numbers. It considers the ratio of successive terms of the series; if the ratio tends to a limit less than 1 for terms further along the series, then it converges absolutely.

Related Topics:
Mathematics - Series - Absolutely

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The test was first published by Jean le Rond d'Alembert and is sometimes known as

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d'Alembert's ratio test.

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Formally, the ratio test states that if

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:lim_{n ightarrowinfty}left|rac{a_{n+1}}{a_n} ight|

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then the series

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:sum_{n=1}^infty a_n

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converges absolutely, and if

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:lim_{n ightarrowinfty}left|rac{a_{n+1}}{a_n} ight|>1

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then the series diverges. In particular, if

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:lim_{n ightarrowinfty}left|rac{a_{n+1}}{a_n} ight|

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exists, then the series converges absolutely if that limit is < 1 and diverges if it is > 1. There exist both convergent and divergent series for which the limit is exactly 1, and consequently the test is inconclusive in that case.

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An extension of the ratio test due to Raabe allows one to sometimes deal with the case when the limit is exactly 1. Raabe's test states that

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if

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:lim_{n ightarrowinfty}left|rac{a_{n+1}}{a_n} ight|=1

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and if a positive number c exists such that

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:lim_{n ightarrowinfty}

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,nleft(,left|rac{a_{n+1}}{a_n} ight|-1 ight)=-1-c

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then the series will be absolutely convergent. d'Alembert's test and Raabe's test

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are the first and second theorem in a hierarchy of such theorems due to

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Augustus De Morgan.

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