Answer
Divergent
Work Step by Step
Alternating series test:
Suppose that we have series $\Sigma a_n$, such that $a_{n}=(-1)^{n}b_n$ or $a_{n}=(-1)^{n+1}b_n$, where $b_n\geq 0$ for all $n$.
Then if the following two condition are satisfied the series is convergent.
1. $\lim\limits_{n \to \infty}b_{n}=0$
2. $b_{n}$ is a decreasing sequence.
Given: $-\frac{2}{5}+\frac{4}{6}-\frac{6}{7}+\frac{8}{8}-\frac{10}{9}+....$
The terms are becoming larger, $\frac{2}{5}\lt \frac{4}{6} \lt \frac{6}{7} \lt \frac{8}{8} \lt\frac{10}{9}+....$
Thus, Alternating Series Test does not apply.
We can come up with the formula that generates the terms:
General Term $\Sigma _{n=1}^{\infty}(-1)^{n}\frac{2n}{n+4}$
Evaluate as $n$ approaches infinity.
Thus, $-\frac{2}{5}+\frac{4}{6}-\frac{6}{7}+\frac{8}{8}-\frac{10}{9}+....=\Sigma_{n=1}^{\infty}(-1)^{n}\frac{2n}{n+4}$
$=\Sigma_{n=1}^{\infty}(-1)^{n}\frac{2}{1+\frac{4}{n}}$
$=\Sigma_{n=1}^{\infty}(-1)^{n}(2)= DNE$
which means that the limit does not exist, so the series diverges by the Test of Divergence.
