Testing Divergence: Alternating Series

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The discussion centers on the application of the Test for Divergence to alternating series. It clarifies that while the limit of an alternating term like (-1)^n does not exist, this alone does not indicate divergence. The example of the series ((-1)^n)/n illustrates that despite the alternating nature, the limit approaches zero, allowing for convergence. Thus, the divergence test does not apply effectively in this context, as it fails to provide conclusive information about convergence or divergence. Ultimately, the alternating series can converge despite the divergence test's initial implications.
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Can the Test for Divergence (limit of an->infinity not equal to zero) be used on an alternating series?
For example, if a series has a (-1)^n term. Can we assume that since the limit of that term does not exist, then the series is automatically diverging?
 
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Yes.
 
Ok,
but what about the sum of ((-1)^n)/n? Doesn't the divergence test say that this sum diverges because of the alternating 1, while the series converges with the alternating series test..
 
No, because

<br /> \lim_{n \to \infty} \frac{(-1)^n}{n} = 0<br />

so that test doesn't provide any information.
 
oh alright! thank you
 

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