Determine whether the series is convergent or divergent. P series. Can I use the integral test too?

Question

Say I have this series:

$$\sum_{n=1}^{+\infty} \frac{1}{n^2+4}$$

So I know I can just use a p-series test here. The above fraction is $< \frac{1}{n^2}$ which converges. So the above series must converge too. is there a way to show this converges by the Integral Test too? I can't u-sub since there isn't an n in the numerator to usub with...

Here is my p-series definition:

Answer 1

Yes, you can. Just use the fact that$$\int\frac{\mathrm dx}{x^2+4}=\frac12\arctan\left(\frac x2\right).$$