Evaluating summation of arctan function

Question

How should I proceed in evaluating the following:

$$ \sum_{r=1}^{\infty} \arctan\left(\frac{2}{r^2+r+4} \right)$$

I thought the denominator would be like: $(x+a)(x+a+2)+1$, but it didn't worked out. Any hints?

Answer 1

Hint

As the denominator needs something of the form $$1+f(r)f(r+1)$$

where $f(r)=ar+b$ with $a,b$ arbitrary constants

$$\dfrac2{r^2+r+4}=\cdots=\dfrac{(r+1)/2-r/2}{1+r/2\cdot(r+1)/2}$$