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If r > 1, then the series diverges. If r = 1, the root test is inconclusive, and the series may converge or diverge. The root test is stronger than the ratio test: whenever the ratio test determines the convergence or divergence of an infinite series, the root test does too, but not conversely. [1]
In mathematics, the limit comparison test (LCT) (in contrast with the related direct comparison test) is a method of testing for the convergence of an infinite series. Statement [ edit ]
In mathematics, the comparison test, sometimes called the direct comparison test to distinguish it from similar related tests (especially the limit comparison test), provides a way of deducing whether an infinite series or an improper integral converges or diverges by comparing the series or integral to one whose convergence properties are known.
The more general class of p-series, =, exemplifies the possible results of the test: If p ≤ 0, then the nth-term test identifies the series as divergent. If 0 < p ≤ 1, then the nth-term test is inconclusive, but the series is divergent by the integral test for convergence.
Get ready for all of today's NYT 'Connections’ hints and answers for #542 on Wednesday, December 4, 2024. Today's NYT Connections puzzle for Wednesday, December 4, 2024 The New York Times
Here the series definitely converges for a > 1, and diverges for a < 1. When a = 1, the condensation transformation gives the series (). The logarithms "shift to the left". So when a = 1, we have convergence for b > 1, divergence for b < 1. When b = 1 the value of c enters.
Get ready for all of today's NYT 'Connections’ hints and answers for #578 on Thursday, January 9, 2025. Today's NYT Connections puzzle for Thursday, January 9, 2025 The New York Times
Get ready for all of today's NYT 'Connections’ hints and answers for #489 on Saturday, October 12, 2024. Today's NYT Connections puzzle for Saturday, October 12, 2024 The New York Times