<s>[INST] 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.
 [/INST] You want to use a comparison test. Remember with series that you can disregard a finite number of terms. When $n \geq 2$
$$n^7+13n^5+9n+2 \leq n^7 +13n^7 + 9n^7+n^2 = 24n^7,$$ 
which implies
$$ \frac{1}{n^7+13n^5+9n+2} \geq \frac{1}{n^7 +13n^7 + 9n^7+n^2} \geq \frac{1}{24n^7}.$$ 
For the numerator, you have 
$$n^6 + 13n^5 + n + 1 \geq n^6.$$
Therefore
$$ \sum_{n=2}^\infty \frac{n^6 + 13n^5 + n + 1}{n^6 + 13n^5 + 9n + 2} \geq \frac{1}{24} \sum_{n=2}^\infty \frac{n^6}{n^7} = \frac{1}{24}\sum_{n=2}^\infty \frac{1}{n} =\infty, $$
which shows that the series diverges by comparison with the harmonic series. 
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