1. Nov 5, 2007

### frasifrasi

How can I show that the series from 1 to infinity of

(-1)^(n+1) * n / sqrt(n^(2)+2) diverges instead of converging abs/conditionally?

Also, for the series from 1 to infinity of:

(-1)^(n+1)/(2)^(1/n)

I applied the root test and came out with:

lim n --> infinity of 1^n/2 = 1/2

yet, the answer key says the series diverges...can anyone explain this?

Last edited: Nov 5, 2007
2. Nov 5, 2007

### Dick

The nth term of either of those series doesn't even converge to zero.

3. Nov 5, 2007

### Antineutron

try the alternating series test.

4. Nov 5, 2007

### Antineutron

if you just try to simplify and do the limits the denomintor goes to 1 and the top alternates from -1 to 1 and back and forth.

5. Nov 5, 2007

### Dick

Well, yeah. Isn't that what I said?

6. Nov 5, 2007

### frasifrasi

I see, the limit of 1/(2)^1/n goes to 1, so it fails to go to 0 and hence diverges.

and the first can be simplified to n/n = 1, which doesn't go to 0, so it diverges...

Is it that simple?

7. Nov 5, 2007

### Dick

It is that simple. When are applying a test like the alternating series test, make sure that ALL of the premises apply. Of course, just because a test doesn't apply, doesn't make the series diverge. But any series that for large n looks like +1,-1,+1,-1... does not converge.

Last edited: Nov 6, 2007