Converges or Diverges

**kl.twilleger** · November 13th 2008, 12:01 PM

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

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

**Mathstud28** · November 13th 2008, 12:25 PM

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 $\int_2^{\infty}\frac{dx}{x\ln(x)}$

**albi** · November 13th 2008, 01:00 PM

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.

**Mathstud28** · November 13th 2008, 01:03 PM

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.

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