RATIO TEST Meaning and
Definition
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The ratio test is a mathematical tool used to determine the convergence or divergence of an infinite series. It is a method employed to analyze the behavior of a series by examining the ratio between consecutive terms. The ratio test is particularly useful when dealing with series whose terms involve exponentiation or factorials.
To apply the ratio test, consider an infinite series ∑ aₙ, where aₙ represents the terms of the series. The ratio test states that if the limit as n approaches infinity of the absolute value of (aₙ₊₁ / aₙ) is less than 1, then the series converges absolutely. This means that the series will approach a finite sum as more terms are added, and the sum will not be influenced by the order in which the terms are added. Conversely, if the limit is greater than 1 or undefined, the series diverges, meaning that it does not approach a finite sum.
In other words, the ratio test determines whether a series is convergent or divergent based on the rate at which its terms increase or decrease. It establishes the conditions under which an infinite series will exhibit either a bounded sum or unbounded growth. The ratio test is a powerful tool in mathematical analysis as it enables mathematicians to ascertain the convergence properties of a wide array of series.
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Etymology of RATIO TEST
The word "ratio" comes from the Latin word "ratiō", which means "reckoning, reasoning, calculation".
The term "ratio test" is a mathematical concept that was introduced by the Swiss mathematician Leonhard Euler in the 18th century. It is used to determine the convergence or divergence of an infinite series.
The use of the word "ratio" in the term "ratio test" refers to the division or comparison of successive terms in the series. By examining the ratio between consecutive terms, the ratio test helps to analyze the behavior of the series, whether it converges to a finite limit, diverges to infinity, or does not converge at all.
Similar spelling words for RATIO TEST
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