Series converge or diverge?

[zeion]

Hi can I get some help on this

 

The problem statement, all variables and given/known data

 

Determine whether the series converges or diverges.

 

[math] \sum \frac{lnk}{k^3} [/math]

 

 

 

 

 

 

The attempt at a solution

 

Since lnk always less than 0, so [math] \frac{lnk}{k^3} \leq \frac{1}{k^3}[/math] and [math]\frac{1}{k^3} [/math]diverges so[math] \frac{lnk}{k^3}[/math] diverges.

[lawcat]

The divergence test is done by taking the limit of the main term. In your case, you have to take the limit of [ ln K / K^3.] To do that , you have to apply L'Hospital's rule, and differentiate numerator and denominator. In that case, you will get the limit of [1/3K^3]. If that limit is 0, then the series is converging. If not, then it diverges.

[A23]

The criterion above is not sufficient, for your case it is ok, but for example the series :[math]\sum_k\frac{1}{k}[/math] diverges even if the term tends towards 0.

 

You have comparison criterion or integrating instead of summing, but i do not have here the condition of applicability.