Q:

Prove the series $\sum_{n=1}^∞ (-1)^n(n)/(n+2)$ diverges

First I tried using the ratio test but that did not work because it was inconclusive. I think I have to use the alternating series test. If I can prove that the underlying sequence diverges then I can then say that the series diverges, however I do not know where to begin to show that the sequence $(-1)^n(n)/(n+2)$ diverges. Any tips on what methods to use would be appreciated. Thanks.

A:

The nth-term test says that if $\displaystyle\lim_{n\to\infty} a_n\neq 0$ or this limits does not exist, then $\sum_{n=1}^\infty a_n$ diverges. In your case one can factor out $n$ of the absolute value and see that  $$\frac{n}{n+2}=\frac{1}{1+2/n}\overset{n\to\infty}{\longrightarrow}1\neq 0.$$
Hence
$$\sum_{n=1}^\infty\frac{n}{n+2}$$
diverges and thus
$$\sum_{n=1}^\infty(-1)^n\frac{n}{n+2}$$
as well (in accordance to the alternating series test).

