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alexmahone
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Given a power series, what is the condition on its coefficients that means the ratio test can be applied?
Alexmahone said:Given a power series, what is the condition on its coefficients that means the ratio test can be applied?
Alexmahone said:Given a power series, what is the condition on its coefficients that means the ratio test can be applied?
The ratio test is a method used to determine the convergence or divergence of a power series. It involves taking the limit of the absolute value of the ratio of consecutive terms in the series. If this limit is less than 1, the series is convergent. If it is greater than 1, the series is divergent. If the limit is exactly 1, the test is inconclusive and another method must be used.
The ratio test is typically used when the terms of a power series do not have a clear pattern or when other tests, such as the comparison test, do not apply. It is also useful for determining the radius of convergence of a power series.
To apply the ratio test, you first take the absolute value of the ratio of consecutive terms in the series. Then, take the limit of this value as n approaches infinity. If the limit is less than 1, the series is convergent. If it is greater than 1, the series is divergent. If the limit is exactly 1, the test is inconclusive and another method must be used.
The ratio test and the root test are both methods used to determine the convergence or divergence of a series. The main difference between them is that the ratio test compares consecutive terms in the series, while the root test takes the nth root of the absolute value of each term. Additionally, the ratio test can be used to determine the radius of convergence, while the root test cannot.
No, the ratio test can only be used for power series with non-negative terms. If the terms in the series have a clear pattern or the series is a geometric series, other tests such as the divergence test or the geometric series test should be used instead.