hahaha158 said:Homework Statement
I have a series
Ʃ(1 to infinity) ((-1)^n*n^n)/n!
Homework Equations
The Attempt at a Solution
apparently you cannot use the alternating series for this question, why is this? It has the (-1)^n, what else is needed to allow you to use the alternating series test?
Dick said:To apply the alternating series test you also need that the absolute value of the terms is decreasing. Yours aren't. E.g. (-1)^n*n doesn't converge.
hahaha158 said:Would that not fall under one of the rules for alternating series (lim An as n->∞ must equal to 0 for the series to be convergent)? How can you determine a series to be divergent from this part of the test if you are not even able to apply it?
Dick said:If ##a_n## as a sequence does not converge to zero then the series ##\Sigma a_n## does not converge. Try that test.
Dick said:Yes, the alternating series test can only be applied to test for convergence. Failure of the alternating series test doesn't prove anything. Try another test. As I said, if ##a_n## doesn't converge to zero the series does NOT converge. That's a good first test for divergence.
The alternating series test is not applicable when the series is not alternating. This means that the signs of the terms in the series do not alternate between positive and negative. In this case, the alternating series test cannot be used to determine the convergence or divergence of the series.
No, the alternating series test can only be used for series with monotonically decreasing terms. This means that the absolute values of the terms must decrease as the series progresses. If the terms are not monotonically decreasing, the alternating series test cannot be applied.
Yes, there are several other convergence tests that can be used to determine the convergence or divergence of a series. Some common ones include the ratio test, the root test, and the integral test. It is important to choose the appropriate test based on the characteristics of the series.
No, the alternating series test is not always conclusive. In some cases, the series may satisfy the conditions of the test but still converge or diverge by other means. It is important to use caution and consider other tests or methods when using the alternating series test.
Yes, the alternating series test can be used for both finite and infinite series. However, the series must still satisfy the conditions of the test in order for it to be applicable. It is important to note that the alternating series test is only a sufficient condition for convergence and does not guarantee convergence in all cases.