Is there a more elegant way to prove that $\sum_{n=1}^{\infty}{\frac{n^3+2n^2+1}{n^4-10n-7}}$ diverges? 
Is there a more elegant way to prove that
  $\sum_{n=1}^{\infty}{\frac{n^3+2n^2+1}{n^4-10n-7}}$ diverges?

Here's my (to my mind) messy proof using the comparison test:$$\frac{n^3+2n^2+1}{n^4-10n-7}\geq \frac{n^3}{n^4-10n-7}=\frac{1}{n-\frac{10}{n^2}-\frac{7}{n^3}}\geq \frac{1}{n} \Longleftrightarrow n\geq n-\frac{10}{n^2}-\frac{7}{n^3} \geq 0$$
since the harmonic series diverges, the aforementioned series diverges. Is there a more elegant way? Maybe a neater diverging series to compare it to? I can't use any tests beside the quotient, root and comparsion test.
Especially these rational polynomials really bother me!
 A: The general term of this series is ultimately positive, so one may use equivalents:
A polynomial is asymptotically equivalent to its leading term, so the general term  of your series is equivalent to
$$\frac{n^3+2n^2+1}{n^4-10n-7}\sim_\infty\frac{n^3}{n^4}=\frac1n,$$
which diverges.
A: The quickest :
Near infinity,
$$u_n=\frac{n^3+2n^2+1}{n^4-10n-7} \sim \frac{n^3}{n^4}\sim \frac 1n$$
but 
$$\frac 1n >0$$ and
$$\sum_{n>0} \frac 1n \text{ diverges}$$
thus, the initial series $ \sum u_n $ diverges by limit comparison test for constant sign general term series.
A: For $n>17$, $0<n^4-10n-7<n^4$ and $n^3+2n^2+1>n^3,$ so $$\frac{n^3+2n^2+1}{n^4-10n-7}>\frac{n^3}{n^4}=\frac{1}{n}$$ for $n>17.$
A: Ignore the terms where the denominator is negative. $n(n^{3}+2n^{2}+1) >n^{4}-10n-7$ so the given series dominates $\sum_{n\geq 3} \frac 1 n$ after ignoring $n=1$ and $n=2$. 
A: As said previously by Cheerful Parsnip, there is a test you can apply to this problem: limit comparison test.
Applying it:
Let's compare the original series with $\frac{1}{n}$:
$$\lim_{n\to\infty} \frac{\left(\frac{n^3+2n^2+1}{n^4-10n-7}\right)}{\left(\frac{1}{n}\right)} = \lim_{n\to\infty} \frac{n^4+2n^3+n}{n^4-10n-7}\\$$
Then,
$$\lim_{n\to\infty} \frac{1+\frac{2}{n}+\frac{1}{n^3}}{1-\frac{10}{n^3}-\frac{7}{n^4}} = 1 > 0\\$$
Hence, as the limit tends to 1 when compared to the divergent harmonic series $\sum_{n=1}^\infty \frac{1}{n}$, the original series 
$\sum_{n=1}^\infty \frac{n^3+2n^2+1}{n^4-10n-7}$ also diverges.
If you want more information about it, you can search online or in Simmons George's book Calculus with Analytic Geometry (IIRC).
Be well, my friend!
