Determine if $\sum_{n=1}^\infty \frac{n^2}{n^3+3} $ converges or diverges

Question

Question Determine if the series converges or diverges $$\sum_{n=1}^\infty \frac{n^2}{n^3+3} $$ I tried the nth term divergence test and got 0, which is inconclusive. I also tried comparison test and ratio test. For the ratio test, I got L=1. This looks easy but I just cannot figure out if the series converges or diverges. Any help is much appreciated.

Answer 1

Since $$\frac{n^2}{n^3+3}\geq\frac{1}{n+3}, $$ it diverges.

Answer 2

$\dfrac{n^{2}}{n^{3}+3}\geq\dfrac{n^{2}}{n^{3}+3n^{3}}=\dfrac{1}{4}\dfrac{1}{n}$ and $\displaystyle\sum\dfrac{1}{n}=\infty$.

Answer 3

Since $$\int_{1}^{\infty}\frac{x^2}{x^3+3}dx$$ diverges, the corresponding series does as well by the Integral Test since the integrand is continuous, positive, and decreasing.