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The ratio test is a mathematical test used to determine the convergence or divergence of a series. It involves taking the limit of the ratio of successive terms in the series. If the limit is less than 1, the series is convergent. If the limit is greater than 1 or undefined, the series is divergent.
The ratio test is generally used for series that involve factorials, exponentials, or powers of n. It is also useful for series with alternating signs or alternating terms.
Yes, the ratio test can be used to prove absolute convergence. If the limit of the ratio is less than 1, the series is absolutely convergent, meaning that the series of absolute values converges.
Yes, the ratio test can only be used on series with positive terms. It also may not be conclusive for certain series, such as alternating series or those with terms that approach 1.
The ratio test is considered to be one of the most powerful convergence tests, as it can determine the convergence or divergence of many types of series. However, it may not always be the most efficient method and other tests, such as the comparison test or the integral test, may be more suitable for certain series.