find the values of p for which the series is convergent:
$\displaystyle \sum_{n=3}^{\infty}\frac{1}{n ln n [ln(ln n)]^p}$
please help!
integral test ...
$\displaystyle \int_3^{\infty} \frac{1}{x\ln{x}[\ln(\ln{x})]^p} \, dx$
$\displaystyle u = \ln{x}$
$\displaystyle du = \frac{1}{x} \, dx$
$\displaystyle \int_{\ln{3}}^{\infty} \frac{1}{u(\ln{u})^p} \, du$
consider three cases ... p < 1, p = 1, and p > 1
i'm kind of confused because i thought that the integral test was supposed to test for convergence or divergence..but don't you already know it will be convergent somewhere, because it's a p-series? i thought that you couldn't use the evaluation of the integral to tell you where the series actually converges to.. so, should i just evaluate the limit of the improper integral (once i've integrated) and then.. how do the p values i'm supposed to consider come into play?
if p=1 then i know it just comes out to be ln t...but if it's >1 or < 1 then it's the power rule... so how do you test if it's convergent? i know it's not right to just come on here and ask for answers, but i'm kind of struggling, do you think you could walk me through it?