_ n → ∞ tan \ ∑ _ r = 1 ^ n ^ - 1 ( 1 1 + r + r ^ 2 )\ is equal...

$= \lim_{n \to \infty}\tan \left(\mathop{∑}_{r = 1}^{n}\tan^{- 1}\left(\dfrac{r + 1 - r}{1 + r\left(r + 1\right)}\right)\right)$

$= \tan \left(\lim_{n \to \infty}\mathop{∑}_{r = 1}^{n}\left[\tan^{- 1}\left(r + 1\right) - \tan^{- 1}\left(r\right)\right]\right)$

$= \tan \left(\lim_{n \to \infty}\left(\tan^{- 1}\left(n + 1\right) - \dfrac{\pi}{4}\right)\right)$

$= \tan \left(\dfrac{\pi}{4}\right) = 1$