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? 

As this is a series with positive terms, you may use equivalents: a polynomial function is asymptotically equivalent to its leading term, so
$$\frac{\sqrt{n^3+n+1}}{n^4}\sim_\infty\frac{\sqrt{n^3}}{n^4}=\frac 1{n^{5/2}},$$
which is a $p$-series ($p=5/2$)  converging to $\zeta(5/2)$.
