ratio test

mathematics

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July 13, 2026

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The ratio test is a method used in mathematics to determine if an infinite series converges or diverges. It is a specific application of the comparison test.

The test involves examining the ratio of consecutive terms in a series where all terms are positive. If the limit of this ratio, as n approaches infinity, is less than 1, the series converges. If the limit is greater than 1, the series diverges. If the limit equals 1, the test is inconclusive, and other methods must be used. This test is particularly useful for series involving factorials or exponential terms, as these often simplify when the ratio is taken.1

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infinite series
…reformulated slightly and called the ratio test: if a_n > 0 and if a_{n + 1}/a_n ≤ r for some r < 1 for every n, then a₁ + a₂ +⋯ converges. For example, the ratio test proves the convergence of the series
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…reformulated slightly and called the ratio test: if a_n > 0 and if a_{n + 1}/a_n ≤ r for some r < 1 for every n, then a₁ + a₂ +⋯ converges. For example, the ratio test proves the convergence of the series
Read More