# How do you determine the convergence or divergence of Sigma ((-1)^nsqrtn)/root3n from [1,oo)?

Jan 22, 2017

The series is not convergent as it does not satisfy Cauchy's necessary condition.

#### Explanation:

If we write the general terms of the series as:

${a}_{n} = \frac{\sqrt{n}}{\sqrt[3]{n}} = {n}^{\frac{1}{2}} / {n}^{\frac{1}{3}} = {n}^{\frac{1}{2} - \frac{1}{3}} = {n}^{\frac{1}{6}}$

we can see that:

${\lim}_{n \to \infty} {a}_{n} = \infty$

so the series cannot converge.