The ratio test is a method used to determine whether a series converges or diverges. It involves taking the limit of the absolute value of the ratio of consecutive terms in the series. If the limit is less than 1, the series converges. If the limit is greater than 1 or infinite, the series diverges.
The limit as n approaches infinity in the ratio test represents the behavior of the series as the number of terms increases. It is an important factor in determining whether the series converges or diverges. If the limit is less than 1, the series converges. If the limit is greater than 1 or infinite, the series diverges.
To apply the ratio test to a series, you first find the absolute value of the ratio of consecutive terms in the series. Then, take the limit as n approaches infinity. If the limit is less than 1, the series converges. If the limit is greater than 1 or infinite, the series diverges. If the limit is exactly 1, the test is inconclusive and another method must be used to determine convergence or divergence.
The ratio test can only determine absolute convergence. Absolute convergence means that the series converges regardless of the order in which the terms are added. The test cannot determine conditional convergence, which means that the series converges only when the terms are added in a specific order.
The ratio test is a more general test than the comparison and limit comparison tests. It can be used to determine convergence or divergence of series that cannot be evaluated using these other tests. However, for some series, the ratio test may yield an inconclusive result, in which case another test must be used to determine convergence or divergence.