Convergence of the alternating series (-1)^(n-1)(2n+1)/(n+2)

[SigurRos]

I apologize right now for the fact that I have no idea how to use LaTeX
I can't figure out if the following alternating series is convergent or not:
Sum(((-1)^(n-1)) * ((2n+1)/(n+2))) from 1 to infinity
the root test is not applicable, A(n+1)>An, and the ratio test gives me Limit=1, so I have no comclusive evidence either way. Even Maple 10 couldn't give me an answer.
I have a test tomorrow. HELP!

 

[mathman]

Since it is alternating, you can combine terms pairwise. You will then get a monotonic series with terms ~2/n for large n. This is, as you should know, divergent.

 

[NateTG]

$$\sum_1^\infty (-1)^{n-1} \frac{2n+1}{n+2}$$
Is an alternating series. Therefore it converges if, and only if, the limit of the individual terms goes to zero.
$$\lim_{n \rightarrow \infty} (-1)^{n-1} \frac{2n+1}{n+2}$$
does not exist. (There are limit points at +2 and -2.) Since the sequence of terms does not converge, the series cannot converge.