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Bashyboy said:
The Ratio Test is a mathematical test used to determine the convergence or divergence of an infinite series. It involves taking the limit of the ratio of the absolute value of consecutive terms in the series. If the limit is less than 1, the series is convergent. If the limit is greater than 1, the series is divergent. If the limit is equal to 1, the test is inconclusive.
The Ratio Test should be used when the series does not have a clear pattern and the terms do not approach 0 as n approaches infinity. It is especially useful for series with factorials, exponentials, or powers.
To perform the Ratio Test, you must first take the absolute value of the terms in the series. Then, take the limit as n approaches infinity of the ratio of the absolute value of the (n+1)th term to the absolute value of the nth term. If the limit is less than 1, the series converges. If the limit is greater than 1, the series diverges. If the limit is equal to 1, the test is inconclusive.
The Ratio Test and Root Test are both used to determine the convergence or divergence of an infinite series. The main difference between the two is that the Ratio Test involves taking the limit of the ratio of consecutive terms, while the Root Test involves taking the limit of the nth root of the absolute value of the nth term. The Ratio Test is best used for series with factorials, exponentials, or powers, while the Root Test is better for series with nth powers or nth roots.
If the limit in the Ratio Test is inconclusive (equal to 1), then the test cannot determine whether the series converges or diverges. In this case, you may need to use other tests, such as the Comparison Test or the Integral Test, to determine the convergence or divergence of the series.
