Find the positive values of $\displaystyle p$ for which the series converges:
$\displaystyle \sum_{n=2}^{\infty} \frac {1}{n (\ln n)^p}$
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Find the positive values of $\displaystyle p$ for which the series converges:
$\displaystyle \sum_{n=2}^{\infty} \frac {1}{n (\ln n)^p}$
Hello,
Take a look at this : Bertrand Series ;)
$\displaystyle \int_1^{\infty}\frac{1}{n\ln(n)^{p}}dn$=$\displaystyle \frac{(\ln(n))^{-p+1}}{-p+1}+C$ and I think you can see where to go from there...just remmber that it must converge(the integral) for the series to converge..oh yeah and if $\displaystyle p=1$ then the antiderivative is $\displaystyle \ln(\ln(n))$ +C
But you did it from 1 to infinity, and it is 2 to infinity.......
Do it when $\displaystyle n=2$ and apply the integral test.
Lemme check my answer, is it when p > 1?
I think thats right
