Answer
diverges; infinite oscillations
Work Step by Step
To determine if the sequence converges, take the limit as n approaches infinity for the sequence.
$\lim\limits_{n \to \infty} (-1)^{n}(\frac{n}{n+1})$
Since the limit of products is the product of the limits,
$=(\lim\limits_{n \to \infty}(-1)^{n})(\lim\limits_{n \to \infty}\frac {n}{n+1})$
$\lim\limits_{n \to \infty}(-1)^{n}$ oscillates between -1 and 1 infinitely, so no limit can be determined. Also, the limit as n approaches infinity for $\frac {n}{n+1}$ is 1. So, $1\times D.N.E.=D.N.E.$. Therefore, the limit doesn't exist at infinity and the sequence diverges.
