Simplifying $\arctan\frac{2}{1^2} + \arctan\frac{2}{2^2} + \arctan\frac{2}{3^2}+ \cdots+\arctan\frac{2}{n^2}$

Question

$$S = \arctan\frac{2}{1^2} + \arctan\frac{2}{2^2} + \arctan\frac{2}{3^2}+ \cdots+\arctan\frac{2}{n^2}$$

I tried to simplify this summation by substituting $\arctan(2/k^2)$ to $a_k$ so that$\ \tan(a_k) = 2/k^2$, then using the sum of tangent formula, but it didn't work. Can someone help me how to simplify this summation?

Answer 1

Hint:

$$\dfrac2{n^2}=\dfrac{n+1-(n-1)}{1+(n-1)(n+1)}$$