Convergence of $\sum_{n=1}^{\infty} \frac{\ln n }{\sqrt{n^3-n+1}}$ Why the following series
$$\sum_{n=1}^{\infty} \frac{\ln n }{\sqrt{n^3-n+1}}$$
converges? 

Note that by L'Hospital's Rule, or other methods,$\lim_{n\to \infty} \frac{\log n}{n^{1/4}}=0$. So if $n$ is large enough, $\log n \lt n^{1/4}$.
Note also that $n^3-n+1\gt n^3/4$. Thus if $n$ is large enough, then
$$\frac{\log n}{\sqrt{n^3-n+1}}\lt \frac{2}{n^{5/4}}.$$
It follows by Comparison that our series converges. 
