## University Calculus: Early Transcendentals (3rd Edition)

The $n$-th Term Test states that when a series $a_n \to 0$ then the series diverges. We notice that $\lim\limits_{n \to \infty} a_n=\lim\limits_{n \to \infty} (-1)^n \dfrac{n}{n+1}$ a) When $n$ is even, then $\lim\limits_{n \to \infty} a_n=\lim\limits_{n \to \infty} \dfrac{n}{n+1}=\lim\limits_{n \to \infty} \dfrac{n}{1+1/n}=\dfrac{1}{1+0}=1$ b) When $n$ is odd, then $\lim\limits_{n \to \infty} a_n=\lim\limits_{n \to \infty} -\dfrac{n}{n+1}=\lim\limits_{n \to \infty} -\dfrac{n}{1+1/n}=-\dfrac{1}{1+0}=-1$ Therefore, the given series Diverges by the $n$-th Term Test.