$\displaystyle \sum_{n=1}^\infty (-1)^{n-1}\frac{\sqrt{n}}{ n+2 }$
I tried the ratio test and I got 1, which means inconclusive. What can I do to tell whether it is converging or diverging?
Let $\displaystyle a_n=\frac{\sqrt{n}}{n+2}$. Note that $\displaystyle \forall{n}\in\mathbb{N}~a_{n+1}\leqslant{a_n}$. Further note that $\displaystyle \lim_{n\to\infty}a_n=0$. So this series converges by Leibniz's Criterion. Note that although the Ratio Test is very useful, a lot of kids get in a Ratio Test only mindframe, try not to do that.
$\displaystyle \sum_{n=1}^\infty (-1)^n\frac{n}{ 5 + \ln n }$
so this is also alternating. and it looks like it's oscillating. Therefore, it's diverging?
Also
$\displaystyle \sum_{n=1}^\infty (-1)^n\frac{ 4 n }{ 8 n + 7 }$
This one is oscillating. it means diverging, right?
In this case oscillating and alternating are synonomous. And secondly no that is completely incorrect. Holistically being alternating greatly increases the chances a series converges. It mus satisfy much less stringent requirements. Do you know the Alternating Series Test (Leibniz's Criterion)?
The Alternating Series Test (Lebniz's Criterion) loosely states: If $\displaystyle \sum{(-1)^na_n}$ is an infinite series it converges iff $\displaystyle \lim_{n\to\infty}a_n=0$ and $\displaystyle a_n\in\downarrow$ or if you prefer $\displaystyle a_{n+1}\leqslant{a_n}$
Pauls Online Notes : Calculus II - Alternating Series Test
you're not an "online" student, are you?