Determine if this series $ \sum_{n=0}^\infty\frac{n^6+13n^5+n+1}{n^7+13n^4+9n+2}$ converges Determine if the following series converges:
$$
\sum_{n=0}^\infty\frac{n^6+13n^5+n+1}{n^7+13n^4+9n+2}.
$$
(http://i.stack.imgur.com/qWiuy.png)
I don't know how to  start.

Use the limit comparison test. 
Let $\displaystyle a_n=\frac{n^6+13n^5+n+1}{n^7+13n^4+9n+2}$ and $\displaystyle b_n=\frac{n^6}{n^7}=\frac{1}{n}$.
Since 
$$\begin{align}\displaystyle\lim_{n\to\infty}\frac{a_n}{b_n}&=\lim_{n\to\infty}\frac{n^6+13n^5+n+1}{n^7+13n^4+9n+2}\cdot\frac{n}{1}\\
&=\lim_{n\to\infty}\frac{n^7\left(1+13\frac{1}{n}+\frac{1}{n^5}+\frac{1}{n^7}\right)}{n^7\left(1+13\frac{1}{n^3}+9\frac{1}{n^6}+\frac{2}{n^7}\right)} \\
&=1 > 0 \end{align}$$
Both $a_n$ and $b_n$ converges or diverges. Since $b_n$ diverges, so too must $a_n$.
