# How do I test $\sum _ { n \geq 1 } \frac { \sqrt{ n^{3}+5n-1}}{ n^{2} - \sin n^{3}}$ for convergence/divergence?

How do I test $$\sum _ { n \geq 1 }\frac { \sqrt{ n^{3}+5n-1}}{ n^{2} - \sin n^{3}}$$ for convergence/divergence?

I tried comparison tests, but couldn't find any smaller diverging series that I could compare this one to, could not figure out how to do limit tests either.

• Try $a/sqrt(n)$ for some $a$. – Owen May 2 at 16:34

Well you have : $$\dfrac{\sqrt{n^3+5n-1}}{n^2-\sin(n^3)} \sim\dfrac{n^{3/2}}{n^2} \sim \dfrac{1}{n^{1/2}}$$ So as long (1/2<1) you can use Riemann's rule to have your result.