How to analyze (sum and convergence) step-by-step this series: $\sum_{n=1}^\infty\frac{4}{(n+1)(n+2)}$? I have this series:
$$\sum_{n=1}^\infty\frac{4}{(n+1)(n+2)}$$
and $$\sum_{n=1}^\infty\frac{4}{(n+1)(n+3)}$$
and I would like to know how to analyze, very step-by-step.
thanks

One way to show convergence of the first series, is to first note that 
$$ \frac{4}{(n+1)(n+2)} < \frac{4}{(n+1)^2}$$ for all $n$. Then write
$$ \sum_{n = 1}^{\infty} \frac{4}{(n+1)^2} = 4\sum_{n=2}^{\infty} \frac{1}{n^2}. $$ Presumably, you already know that the latter series converges, and so you can apply the comparison test, to show that 
$$ \sum_{n = 1}^{\infty} \frac{4}{(n+1)(n+2)} $$ converges. The other one can be treated in a similar way. 
To find the sum, write 
$$ \frac{1}{(n+1)(n+2)} = \frac{A}{n+1} + \frac{B}{n+2},$$ and solve for $A$ and $B$ (this is called a partial fractions decomposition). Then you should be able to determine the $n$'th partial sum of this series explicitly, and easily evaluate the sum (these types of series will be called telescopic in your textbook, for obvious reasons). The other one can be treated in a similar way, but the computations will be a bit more complicated.
Note: Just realized that the first part of my answer is made redundant by the second, but I'll leave it - it can't hurt to know more than one way of accomplishing something.
Edit: After seeing your comment above, I'll add some more details on the next step too: 
$$ s_k = \sum_{n = 1}^{k} \frac{4}{(n+1)(n+2)} = 4\sum_{n = 1}^{k} \left( \frac{1}{n+1} - \frac{1}{n+2}\right).$$
Let us look more closely at the second sum
$$\sum_{n = 1}^{k} \left( \frac{1}{n+1} - \frac{1}{n+2}\right) = \left( \frac{1}{2} - \frac{1}{3} \right) + \left( \frac{1}{3} - \frac{1}{4} \right) + \left( \frac{1}{4} - \frac{1}{5} \right) + \ldots = \frac{1}{2} - \frac{1}{k+2}.$$ 
So $s_k = 4(1/2 - 1/(k+2))$ and hence the sum of the series is $s = \lim_{k \to \infty} s_k = 4/2 = 2$. 
