Convergence of series with natural logarithm in denominator

[edoz]

Homework Statement

http://img840.imageshack.us/img840/3609/unleddn.png

note that by log(n), i really mean NATURAL log of n

Homework Equations

it's convergent, but I can't figure out which test to use

The Attempt at a Solution

there is no term to the nth power, so ratio test is useless; root test is useless too; comparison test would seem to be the best option, but I can't figure out how to compare when I have a natural log in the expression... limit comparison test takes me nowhere either.

 

[upsidedowntop]

I think that the log(n) is just there to throw you off. $$\log(n) < n^{1/2}$$ asymptotically right. Do you have a theorem showing that $$\sum \frac{1}{n^{3/2}}$$ is convergent?

 

[edoz]

yes, p-series are convergen if p>1;

now i get it, I just need to compare it with the series you just shown, since with the (n+1) the original series will always be smaller than the convergent series you shown, therefore convergent too.