Alternating series test question. I'm reading this in my text:

So the alternating series test says:

i) is about the sequence decreasing
ii) is about the limit of the b term going to 0
I'm confused about why we need to do anything more once we find out that ii) isn't satisfied in example 2. What does it mean that we're looking at the limit of the nth term of the series? Don't we know that it diverges already? Also, can someone show me how they determine the limit of $a_n$?

If conditions (i) and (ii) are satisfied, then you conclude that the series ${\bf converges}$. 
If one of the conditions fails, then you cannot conclude that the series ${\bf diverges}$
The limit $(a_n)$ does not exist because it converges to two different values. In fact, if $n$ is even, then sequence $a_n$ converges to $3/4$ while it converges to $-3/4$ for the odd terms and thus it diverges. In particular, it $\lim a_n \neq 0$ and so series diverges by divergence test.
