How to determine if $$ \sum_{n=0}^{\infty} (-1)^n \frac{1}{n^2 +3n +2}$$ converges or not?
I expand first the series as shown below to observe if I can rewrite it to another simpler form of series but I could not find any way. $$ \frac{1}{2} -\frac{1}{6} + \frac{1}{12} -\frac{1}{20} + \frac{1}{30} -\frac{1}{42}$$
Or is there any faster test for convergence without the need of expanding the series?
It is absolutely convergent because $|\frac {(-1)^{n}} {n^{2}+3n+3}| \leq \frac 1 {n^{2}}$ and $\sum\limits_{n=1}^{\infty} \frac 1 {n^{2}} <\infty$. Note that the term corresponding to $n=0$ can be ignored for determining convergence.