Is series convergent? Does this series converge?  
$$\sum_{n=1}^{\infty }\frac{{(-1)}^{n}(n+2)}{{n}^{2}+4}$$  
How can i step-by-step calculate it?
We have

$$\frac{{(-1)}^{n}(n+2)}{{n}^{2}+4}=\frac{(-1)^n}{n}+O\left(\frac{1}{n^2}\right)$$
and the series $\displaystyle\sum \frac{(-1)^n}{n}$ is convergent by the Leibniz theorem and the series $\displaystyle\sum O\left(\frac{1}{n^2}\right)$ is convergent by comparison with Riemann convergent series hence the given series is convergent.
$ | \frac{n+2}{n^2+4} |$ = $ \frac{n}{n^2+4} + \frac{2}{n^2+4} \leq \frac{1}{n^2} + \frac{2}{n^2} = \frac{3}{n^2}$ this implies $|a_n|$ is convergent therefore the sequence $a_n$ converges. I guess just go about it carefully