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!

If you are looking for the sum of the series
$$
\sum_{n=0}^\infty (\arctan(n+1)-\arctan(n+2))
$$
then what you have is a telescoping series.  This means that terms will cancel.  The first part of the sum is
$$
\arctan(1)-\arctan(2)+\arctan(2)-\arctan(3)+\arctan(3)-\arctan(4)+\cdots
$$
The terms in the middle will cancel, for instance:
$$
-\arctan(2)+\arctan(2)=0
$$
Thus your $n$th partial sum is
$$
S_n=\arctan(1)-\arctan(n+2)
$$
Taking the limit gives
$$
\lim_{n\rightarrow \infty}(\arctan(1)-\arctan(n+2))=\arctan(1)-\frac{\pi}{2}.
$$
And $\arctan(1)=\frac{\pi}{4}$.
