1. ## Converges or Diverges

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

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

2. Originally Posted by kl.twilleger
$\sum_{k=2}^\infty \frac{4}{k ln k}$

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

3. There is an alternative solution to this.

Theorem. If $a_1 > a_2 > ... > 0$ then series $\sum_{k = 1}^\infty a_k$ converges if and only if $\sum_{k = 1}^\infty 2^k a_{2^k}$ converges.

4. Originally Posted by albi
There is an alternative solution to this.

Theorem. If $a_1 > a_2 > ... > 0$ then series $\sum_{k = 1}^\infty a_k$ converges if and only if $\sum_{k = 1}^\infty 2^k a_{2^k}$ converges.
Note to the poster, this is known as Cauchy's Condensation test and is rarely taught at a Calculus level course. It is useful in rare cases, but when it is applicable it is usually powerful.