<s>[INST] Proof for convergence of this series I want to determine if $\displaystyle\sum_{n=1}^\infty\frac{(n^2+n+1)}{(n^4+2n+2)}$ converges.
The ratio test in this case is inconclusive so i tried to do something like this.
$\displaystyle\sum_{n=1}^\infty\frac{(n^2+n+1)}{(n^4+2n+2)}\leq\sum_{n=1}^\infty\frac{(n^2+n+1)}{(n^4)}$
$\displaystyle\int_{1}^{\infty}\sum_{n=1}^\infty\frac{(n^2+n+1)}{(n^4)}=\frac{1}{3}+\frac{1}{4}+\frac{1}{5}$.
Since the second series converges so does the first one.
Is this valid, or how can I else prove if the first series converges?
 [/INST] Yes, you have
$$0\leq \frac{(n^2+n+1)}{(n^4+2n+2)}\leq \frac{(n^2+n+1)}{(n^4)} = \frac{1}{n^2} + \frac{1}{n^3} + \frac{1}{n^4}.$$
And
$$
\sum\frac{1}{n^2} + \frac{1}{n^3} + \frac{1}{n^4}
$$
is a sum of convergent $p$-series, so it is convergent. Now the original series is convergent by the comparison test.
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