Convergence of Series

**kithy** · May 7th 2008, 09:31 AM

Please help me with this!!!

Determine if the following series converge or diverge.

a) the sum from n=1 to infinity of [n/3n+1]^2n

b) the sum from n=2 to infinity of [1/(n(ln)^1/2)]

I don't know what test to use with this series (comparison, ratio, integral etc)?

thank you for your help

**flyingsquirrel** · May 7th 2008, 09:43 AM

Hi

Originally Posted by kithy

Please help me with this!!!

Determine if the following series converge or diverge.

a) the sum from n=1 to infinity of [n/3n+1]^2n
I don't know what test to use with this series (comparison, ratio, integral etc)?

The comparison test works : $0\leq \frac{n}{3n+1}=\frac{1}{3+\frac{1}{n}}\leq \frac{1}{3}$

b) the sum from n=2 to infinity of [1/(n(ln)^1/2)]

I don't know what test to use with this series (comparison, ratio, integral etc)?

You can use integrals : $\int \frac{1}{t \sqrt{\ln t}} \,\mathrm{d}t=2\int \frac{u'(t)}{2\sqrt{u(t)}} \, \mathrm{d}t$ with $u(t)=\ln t$ can be quite easily evaluated.

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