When can you not apply the alternating series test?

[hahaha158]

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]

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?

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]

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.

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]

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?

If $a_n$ as a sequence does not converge to zero then the series $\Sigma a_n$ does not converge. Try that test.

 

[hahaha158]

If $a_n$ as a sequence does not converge to zero then the series $\Sigma a_n$ does not converge. Try that test.

Is there any way to do it only using the alternating series test?

From what i understand this is the alternating series test

Consider Ʃ(n=1 to ∞) (-1)^(n-1)bn where bn>=0. Suppose
I) b(n+1)<= bn eventually
II) lim as n ->∞ bn=0
Then Ʃ(n=1 to ∞) (-1)^(n-1)bn converges.

It seems like what you are saying is that you cannot test for divergence using the alternating series test, only convergence?

The reason i am asking this is beccause of this

http://imgur.com/kTd1Lbt

It has a choice where it can diverge and a choice where you can not apply alternating series test.

So is it true that for an alternating series test you can test for convergence, but if it does not follow the 2 conditions above, then you must take bn (or an) and apply it to a different test to see it if diverges or converges?

 

[Dick]

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.

 

[hahaha158]

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.

ok thanks for clarifying