Hi everyone. Today we are asked to do determine if this infinite series, $\displaystyle \sum_{n=1}^{\infty}\left [ \arctan (x+2)-\arctan(x) \right ]$, converges or diverges and if it converges, to find it's value. Divergence test does not help in determining the series' behavior, so what I did is I did the integral test since the sequence can be represented as a continuous, positive, and decreasing function at the interval $\displaystyle [1,\infty)$. Turns out this series is convergent and I have to evaluate this ugly series. By the looks of it, it looks like a telescopic series so by making a diagram of the partial sums up to n=n, everything cancels out except $\displaystyle \left [ -\arctan(1)-\arctan(2) \right ]+\left [ \arctan(n+1)+\arctan(n+2) \right ]$. So I've figured that taking the $\displaystyle \lim_{n\to{\infty}}[\left [ -\arctan(1)-\arctan(2) \right ]+\left [ \arctan(n+1)+\arctan(n+2) \right ]]$ would give me the value for the series. So I ended up with $\displaystyle \sum_{n=1}^{\infty}\left [ \arctan (x+2)-\arctan(x) \right ]=\frac{3\pi}{4}-\arctan(2)$. Is this solution correct? What do you guys think? Thanks everyone for your help!