# A Question about the Alternating Series Test

1. Apr 7, 2011

### mscudder3

The definition I am working from is "Let Z=(z(sub n)) be a decreasing sequence of strictly positive numbers with lim(Z)=0. Then the alternating series, Sum(((-1)^n)*Z) is convergent.

My question is how to solve the following:
If the hypothesis that Z is decreasing is dropped, show the Alternating Series Test may fail.

I am aware of a proof utilizing some Z that is also alternating, but this breaks the condition that Z is strictly positive. I am unaware of an such sequence that has a limit of 0, all elements of the series are positive, yet is divergent.

2. Apr 8, 2011

### Stephen Tashi

Look at a sequence whose odd terms are a convergent sequence of positive terms, like a geometric series and whose even terms are the terms of the divergent harmonic series {1/n}. The alternating sequence of negative odd terms and positive even terms should diverge. Intuitively this is because what is begin taken away reaches a limit while what is being added doesn't.

It will be a slight nuisance to get the indexing written correctly. I think harmonic terms will be $$\frac{1}{ (n/2)+1}$$ , for $$n = 0,2,4,...$$

3. Apr 8, 2011

### mscudder3

Thanks!

I never really assessed the thought of a limit being reached while another series (running in parallel) continued decreasing. I appreciate the quick input!