What is correct approach for test of divergence/convergence for $\sum_{n=1}^{\infty} \frac{\sqrt{n^3+n+1}}{n^4}$ What is the correct approach for determining if the following series is convergent or divergent? 
$$\sum_{n=1}^{\infty} \frac{\sqrt{n^3+n+1}}{n^4}$$
My thought process was to use the limit comparison test where: 
$$\sum_{n=1}^{\infty} \frac{\sqrt{n^3+n+1}}{n^4} \leq \frac{1}{n^4}$$
According the p-series test this series converges. Is this correct? 

Hint: use the ratio test with ${1\over {n^{5\over 2}}}$
${{\sqrt{n^3+n+1}\over n^4}}$
$={{n^{3\over 2}\over n^4}}
{{\sqrt{1+{1\over n^2}+{1\over n^3}}}}$
$={1\over {n^{5\over 2}}}{{\sqrt{1+{1\over n^2}+{1\over n^3}}}}$
