Chapter 11 - Infinite Series - 11.3 Convergence of Series with Positive Terms - Exercises - Page 556: 15

Converges

Given $$\sum_{n=1}^{\infty} \frac{1}{n^{3}+8 n}$$ We compare the given series with the series $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^{3}}$ $p-$series , $p=3$ convergent and for $n\geq 1$ $$\frac{1}{n^{3}+8 n}\leq\frac{1}{n^{3}}$$ Then $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n^{3}+8 n}$ also converges