I was provided the following first terms of a series in a question from my math book:
$-\frac{2}{5}+\frac{4}{6}-\frac{6}{7}+\frac{8}{8}-\frac{10}{9}+...$
I concluded that this is the equivalent of the following sum representation:
$\sum\limits_{n=1}^\infty(-1)^n\frac{2n}{n+4}$
I then proceeded to do the alternating series convergence test by first comparing two following terms:
$b_n=\frac{2n}{n+4}$
$b_{n+1}=\frac{2n+2}{n+5}$
I concluded that $b_{n+1}>b_n$ for at least one $n$ and thus the series is divergent. The limit test is also conclusive:
$\lim_{n->\infty} \frac{2n}{n+4} = \lim_{n->\infty} \frac{2}{1+\frac{4}{n}} = 2$
Since the limit is not equal to $0$, both alternating series tests point out that the series is in fact divergent.
Unfortunately the answer pages in my math book itself say that the series is convergent. Although Wolfram Alpha confirms my thoughts by saying the series is divergent, it shows no proof or method to reach this answer.
Have I done everything right and is the series in fact divergent?