$\displaystyle \sum_{k=2}^\infty \frac{4}{k ln k}$

should I turn the ln into an e? What rule should I use??

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- Nov 13th 2008, 12:01 PMkl.twillegerConverges or Diverges
$\displaystyle \sum_{k=2}^\infty \frac{4}{k ln k}$

should I turn the ln into an e? What rule should I use?? - Nov 13th 2008, 12:25 PMMathstud28
Let $\displaystyle a_k=\frac{4}{k\ln(k)}$. Consider that $\displaystyle \forall{k}\in[2,\infty)~a_k>0\wedge{a_k}\in\mathcal{C}\wedge{a_k }\in\downarrow$ so the integral test applies. So this series shares convergence/divergence with $\displaystyle \int_2^{\infty}\frac{dx}{x\ln(x)}$

- Nov 13th 2008, 01:00 PMalbi
There is an alternative solution to this.

Theorem. If $\displaystyle a_1 > a_2 > ... > 0$ then series $\displaystyle \sum_{k = 1}^\infty a_k$ converges if and only if $\displaystyle \sum_{k = 1}^\infty 2^k a_{2^k}$ converges. - Nov 13th 2008, 01:03 PMMathstud28