Using ratio test to test conditional convergence?

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The ratio test indicates that if the limit is greater than 1, the series diverges, and if it's less than 1, the series converges absolutely. When the limit equals 1, the test is inconclusive, leaving uncertainty about convergence. Conditional convergence can occur when the ratio test yields a limit of 1, but this is not guaranteed. Additionally, there are cases where the ratio test may not provide a limit, allowing for various convergence behaviors. Therefore, not all conditionally convergent series will necessarily have a limit of 1 when using the ratio test.
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Homework Statement


So what I was taught was that if the lim of the ratio test is the series is always absolutely convergent. If it is >1 the series is always divergent. But if it is =1 then we don't know. So would that mean that all conditionally convergent series would have a limit = 1? I am confused :s

Thanks!
 
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Yes, if the ratio test applied to conditional convergent series exists, then it must be 1. On the other hand, the ratio test might not give you a limit at all. In that case the series can do anything it wants to.
 

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