_ n → ∞ [π n^ 2 + 1 ] is equal to

$\lim_{n \to \infty}\sin \left\{n\pi\left(1 + \dfrac{1}{n2}\right)^{1} / 2\right\}$

$= \lim_{n \to \infty}\sin \left\{n\pi\left(1 + \dfrac{1}{2n^{2}} - \dfrac{1}{8n^{4}} + \ldots\right)\right\}$

$= \lim_{n \to \infty}\sin \left\{n\pi + \dfrac{\pi}{2n} - \dfrac{\pi}{8n^{3}} + \ldots\right\}$

$= \lim_{n \to \infty}\left(- 1\right)nsin\pi\left(\dfrac{1}{2n} - \dfrac{1}{8n^{3}} + \ldots\right) = 0$