prove that sequence $\frac{n^2+1}{n+4}$ diverges Can somebody help me prove this sequence? 

I've tried using Comparison Theorem and ended up with 

$a_n = \frac{n^2+1}{n+4} &lt; \frac{n^2}n = n &gt; N = M$.

So I choose $N = M$.

But I don't think this is correct.
Hint:

$$\frac{n^2+1}{n+4} \geq \frac{n^2+1}{n+n} &gt; \frac{n^2}{2n}=\frac{n}{2}.$$
The first inequality holds for $n \geq 4$. Also we know the behavior of $\left\{\frac{n}{2}\right\}$.
You could use L'Hôpital's rule:   $\lim_{n\to \infty}\dfrac {n^2+1}{n+4}\stackrel{L'h}=\lim_{n\to\infty}\dfrac {2n}1=\infty$.