Converging vs Diverging Series: Understanding the Root Test

[frasifrasi]

URGENT - Please help - easy series

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?

 

[Dick]

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

 

[Antineutron]

try the alternating series test.

 

[Antineutron]

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

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.

 

[Dick]

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.

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

 

[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?

 

[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.