Sum of series: arctan(n+1) - arctan(n+2) where n varies from 0 to infinity arctan(n+1) - arctan(n+2) where n varies from 0 to infinity
I'm having trouble figuring out this problem. I've calculated the first 5 terms of the series and ended up with the following:
-.0.32175 - 0.14189 - 0.07677 - 0.07677 - 0.04758 ......
To me, it is apparent that the series is converging on some number (perhaps -1?) however, I'm not sure how to prove this.
Any help is appreciated!

Of course, what makes this (clearly telescoping) series interesting is that it can be written in not-so-obviously telescoping form; using the formula $\mathrm{arctan}(u)$ + $\mathrm{arctan}(v)$ = $\mathrm{arctan}\bigl({u+v\over 1-uv}\bigr)$, we can write it as $\sum_{n=0}^{\infty}\mathrm{arctan}{1\over (n+1)(n+2)-1}$ — or, shifting indices by one, $\sum_{n=1}^{\infty}\mathrm{arctan}{1\over n^2+n-1}$.
