Testing Divergence: Alternating Series

In summary, the Test for Divergence can be used on an alternating series, but the limit of the alternating term must not equal to zero. However, the sum of ((-1)^n)/n would not be affected by this test as the limit of the term is equal to zero.
  • #1
Sahara
8
0
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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  • #2
Yes.
 
  • #3
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..
 
  • #4
No, because

[tex]
\lim_{n \to \infty} \frac{(-1)^n}{n} = 0
[/tex]

so that test doesn't provide any information.
 
  • #5
oh alright! thank you
 

FAQ: Testing Divergence: Alternating Series

1. What is an alternating series?

An alternating series is a mathematical series in which the terms alternate in sign, meaning every other term is positive and negative. It can be written as a sum of terms with alternating signs, such as 1 - 2 + 3 - 4 + 5 - ...

2. How do you test for divergence in an alternating series?

To test for divergence in an alternating series, you can use the Alternating Series Test. This test states that if the terms of an alternating series decrease in absolute value and approach zero, the series will converge. If the terms do not approach zero, the series will diverge.

3. What is the limit comparison test for alternating series?

The limit comparison test for alternating series is a method for determining convergence or divergence of a series by comparing it to a known series. If the limit of the ratio of the terms of the two series is a finite non-zero number, the series will have the same behavior as the known series.

4. Can an alternating series converge even if its terms don't approach zero?

Yes, an alternating series can still converge even if its terms do not approach zero. This is because the Alternating Series Test only requires that the terms decrease in absolute value, not necessarily approach zero. As long as the terms eventually become small enough, the series can still converge.

5. What is the difference between absolute and conditional convergence in an alternating series?

Absolute convergence in an alternating series means that the series converges when all terms are converted to their absolute values. Conditional convergence means that the series only converges when the terms are not converted to their absolute values. In other words, absolute convergence is a stronger condition for convergence than conditional convergence.

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