$\displaystyle \sum (2n+3)(-1)^n/(7n+11)$
Because it is neither decreasing, nor does it have a limit of 0?
see the complete explanation in this link ...
http://www.math.ucdavis.edu/~saito/courses/21C.s09/alternatingseriestest.pdf
So it's convergent? I was told it was divergent, and I am just here to double check...
Wouldn't you just look at everything but the (-1)^n part, and see that you have $\displaystyle (2n+3)/(7n+11)$ and because the powers of n are the same, the limit will be the ratio of the terms, which is 2/7, which is not 0, therefore it can't converge?