The convergence of the series $\sum (-1)^n \frac{n}{n+1}$ and the value of its sum This sum seems convergent, but how to find its precise  value? 
$$\sum\limits_{n=1}^{\infty}{(-1)^{n+1} \frac{n}{n+1}} = \frac{1}{2}-\frac{2}{3}+\frac{3}{4}-\frac{4}{5}+...=-0.3068... $$ Any help would be much appreciated. 

Be a little more careful. You can see that the limit of each term goes to $1$. With the alternator thrown in there, you essentially have the sum $$1-1+1-1\ldots$$ for large enough $n$. For a sum to be convergent you need $$\lim_{n \to \infty}a_n = 0$$ 
