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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.

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  • $\begingroup$ Try $a/sqrt(n)$ for some $a$. $\endgroup$
    – Owen
    May 2, 2020 at 16:34

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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.

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