Is the Alternating Series Test Appropriate for this Problem?

[cue928]

I am looking at the following problem:
sum from n=1 to inf: (-1)^n * n/(n+2)
I used the alternating series test on this, with an = n/n+2 and a(n+1)=(n+1)/(n+3)
lim an = 1, not equal to zero, so I had the series diverging.

The book simply said lim n-> inf of |an| =1

Is my way right? If not, what should have told me that I could not use the alternating series test?

 

[henry_m]

You seem to have it pretty much right, but be careful:

For a series to converge, it is necessary that the terms tend to zero. This is the argument that the book was using to show the series does not converge. This is not, however, sufficient as is shown by the series \sum 1/n. You can use it to prove a series diverges, but not that a series converges.

It is sufficient for the terms to alternate in sign, decrease monotonically and tend to zero. This is the alternating series test. This isn't necessary though; either of the first two conditions can be violated and a series still converges. You can use the test to prove a series converges, but not that a series diverges, which is what you seem to be trying to do.

 

[cue928]

I think I understand what you are saying but I'm still confused on one part. How do you know when to just say lim n to inf |an|=0 or use the alt. series test?

 

[henry_m]

It's a question of logic; the conclusions of the two tests are different. Here's a flippant example which will hopefully make it clear.

Suppose we have some animal in front of us, and we're trying to work out if it's a dog.

One test we could apply is the "tail test": we see if the animal has a tail. If it doesn't we know it isn't a dog. If it does, we're still not sure: it could be a cat! To use the language I was using before, it is necessary for a dog to have a tail, but not sufficient. Or to put it another way "no tail" implies "not a dog". (And no questions about dogs that lost their tails in unfortunate accidents please ).

Another test we could apply is the "terrier test": we see if it's a terrier. If it is, then it's a dog. But if it isn't, we've reached no conclusions: it could be a spaniel! So being a terrier is a sufficient condition to be a dog, but not a necessary condition. "Terrier" implies "dog".

Now if you take the word "dog" and exchange it for "convergent series", the "tail test" is like checking that the terms tend to zero, and the "terrier test" is like the alternating series test. It turns out that your example isn't a dog, but you are trying to use the fact that it isn't a terrier to prove it, which is no good.

In some circumstances one test will be useful, and in other circumstances the other will be useful. Sometimes, neither of them will be useful and you'll have to use some other test.

Hope that's sort of clear...

 

[Dick]

I am looking at the following problem:
sum from n=1 to inf: (-1)^n * n/(n+2)
I used the alternating series test on this, with an = n/n+2 and a(n+1)=(n+1)/(n+3)
lim an = 1, not equal to zero, so I had the series diverging.

The book simply said lim n-> inf of |an| =1

Is my way right? If not, what should have told me that I could not use the alternating series test?

Do know what the 'alternating series test' is? From what you wrote I'm guessing that what you did is a 'ratio test'. But if lim |an| is not zero, then you don't need any further tests. The sum does not converge. You should always check that first.