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

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Answer 1 · 2017-01-22T21:22:26

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

If we write the general terms of the series as:

$a_n = sqrt(n)/root(3)n = n^(1/2)/n^(1/3) = n^(1/2-1/3) = n^(1/6)$

we can see that:

$lim_(n->oo) a_n = oo$

so the series cannot converge.

How do you determine the convergence or divergence of Sigma ((-1)^nsqrtn)/root3nSigma ((-1)^nsqrtn)/root3n from [1,oo)[1,oo)?