Calculus (3rd Edition)

Given $$\sum_{n=1}^{\infty} \frac{(-1)^{n}}{\sqrt[3]{n}+2 n}$$ Since $$\sum_{n=1}^{\infty}\left|a_{n}\right|=\sum_{n=1}^{\infty} \frac{1}{n^{1 / 3}+2 n}$$ Using the comparison test, we compare with $\sum \frac{1}{n}$, a divergent series: \begin{align*} \lim\limits_{n \to \infty}\frac{n}{n^{1 / 3}+2 n}=1 \end{align*} Hence $\sum_{n=1}^{\infty}\left|a_{n}\right|$ diverges. On the other hand, since $a_n =\dfrac{1}{\sqrt[3]{n}+2 n}$ is decreasing and $\lim_{n\to \infty } a_n=0$, then $\sum_{n=1}^{\infty} \frac{(-1)^{n}}{\sqrt[3]{n}+2 n}$ converges. Hence, $\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n^{1 / 3}+2 n}$ converges conditionally.