# Convergence of Series

• May 7th 2008, 09:31 AM
kithy
Convergence of Series

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)?

• May 7th 2008, 09:43 AM
flyingsquirrel
Hi

Quote:

Originally Posted by kithy

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 : $\displaystyle 0\leq \frac{n}{3n+1}=\frac{1}{3+\frac{1}{n}}\leq \frac{1}{3}$

Quote:

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