sequences and series - When does a sum converge?

I am looking into series and I am stuck at a point when I need to prove convergence of a series, without the convergence tests, and calculate the sum if it converges. I know that a geometric series ($\sum_{n=0}^\infty q^n$) converges when $|q| \le 1$ but I don't know what happens in the case of:
$$\sum_{n=1}^\infty \frac{1}{n(n+2)}$$

for example, or:
$$\sum_{n=1}^\infty \frac{n}{n+1}$$
Can you help me understand what is the actual partition of the series?
Thank you