calculus - Bounding a series: $frac{pi}{2} < sum_{n=0}^infty frac{1}{n^2 + 1} < frac{3pi}{2} $

I have the following statement -

$$\frac{\pi}{2} < \sum_{n=0}^\infty \dfrac{1}{n^2 + 1} < \frac{3\pi}{2} $$

So I tried to prove this statement using the integral test and successfully proved the lower bound. But when I tried to calculate the upper bound I was required to calculate the integral from -1 - $\int_{-1}^{\infty} \frac{1}{x^2 +1}\,dx$.
If someone can explain why it will be great , thanks!