Great! Now I get it

So by using the alternating series test, it can be shown that the series is convergent.

And to test for absolute convergence, using the comparison test:

$\displaystyle \left | a_n \right | \le \left | b_n \right |$

$\displaystyle \sum^\infty_{n=1}\frac{1}{n}\le \sum^\infty_{n=1}(1)^n \left ( \frac{1}{n}+\frac{1}{n-1} \right )$

Since $\displaystyle \sum^\infty_{n=1}\frac{1}{n}$ is divergent, then the absolute of $\displaystyle b_n$ is divergent too, thus it is not absolutely convergent.

So conclusion would be that it's conditionally convergent?

Would the absolute value of the fraction be $\displaystyle \frac{1}{n-1}$ or $\displaystyle \frac{1}{n+1}$? I wasn't quite sure about that.