Question

Determine the convergence or divergence of the series: [(-1)^n+1]/(n+1)

Answer 1

This one is confusing me. I've used the ratio test and determined that L=1, so neither the root or ratio test will help me.

Answer 2

\[\frac{-1^{n+1}}{n+1}\]
this?

Answer 3

Yes. I have to determine the convergence or divergence.

Answer 4

do you know the alternating series test?

Answer 5

I do not. Unfortunately, I was absent for the lesson pertaining to these. I picked up the ratio test on my own, but I don't know any more than that.

Answer 6

http://www.hippocampus.org/HippoCampus/Calculus%20%26%20Advanced%20Math
go to BC calculus and look at the section "alternating series" under sequences and series

Answer 7

That makes sense, thank you. So in simple terms, I would just take the limit of whatever is being multiplied by (-1)^n+1? In this example, that being limit of 1/(n+1)

Answer 8

if it alternates, yes

Answer 9

Would L'Hopital's rule apply at all in problems like this?

Answer 10

I think so

Answer 11

I've never seen a problem like this where it was important though

Answer 12

well, may I give another example?

Answer 13

sure

Answer 14

If I take an alternating series that leaves me with n/(3n+2) as the limit, L'Hopital's rule would give me a limit of 1/3. but as a general rule, wouldn't the fraction be approaching zero because the denominator is getting larger and larger as a faster rate than the numerator? Or would it just approach 1/3 and thus diverge?

Answer 15

it would approach 1/3 and therefore diverge
the denominator is getting larger and larger at a constant multiple of the rate that the numerator is.

Answer 16

Wait, something I just read noted that the alternating series test only tells is a series converges, not if it diverges.