Question

Use root test to determine the following series is convergent/divergent or inconclusive
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Answer 1

By the root test, the series converges. We have

$\displaystyle\lim_{k\to\infty}\sqrt[k]{\left|(-1)^k\frac{k^3}{3^k}\right|}=\lim_{k\to\infty}\frac{\sqrt[k]{k^3}}3=\frac13\lim_{k\to\infty}k^{\frac3k}=\frac13$

which is less than 1.

In case it's not clear why $k^{3/k}$ should converge to 1:

$\displaystyle\lim_{k\to\infty}k^{\frac3k}=\lim_{k\to\infty}\exp\left(\ln k^{\frac3k}\right)=\exp\left(\lim_{k\to\infty}\frac{3\ln k}k\right)\to e^0=1$