Cauchy Test

**MattWT** · December 9th 2010, 06:15 AM

For the series:

$\sum_{n=1}^{\infty} \frac{1}{(n)^(1/2)(logn)^3}$

To show it is divergent/convergent would you first apply the Cauchy test:

$2^n \frac{1}{(2^(n/2) (nlog(2)^)3}$

$\frac{2^(n/2)}{(n^3 log(2)^3)}$

Then apply the ratio test to get the lim = sqrt(2) so it is a divergent series?

Thanks.

**chisigma** · December 9th 2010, 06:32 AM

Is easy to see that exists an $n_{0}$ such that $\forall n>n_{0}$ is...

$\displaystyle \frac{1}{\sqrt{n}\ \ln^{3} n} > \frac{1}{n}$

... and then the series diverges...

Merry Christmas from Italy

$\chi$ $\sigma$

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