$\displaystyle
\sum\limits_{n = 1}^\infty {ln (n) / n^{ p} }
$
I am using integral test as t goes to infinity. I am not familiar with finding the p
Any help.
The sum has to be compared with the integral...
$\displaystyle \int_{1}^{\infty} \frac{\ln x}{x^{p}}\cdot dx$ (1)
Integrating by parts we have...
$\displaystyle \int_{1}^{\infty} \frac{\ln x}{x^{p}}\cdot dx = \frac{1}{-p+1}\cdot \{|x^{-p+1}\cdot \ln x|_{1}^{\infty} - \int_{1}^{\infty} \frac{dx}{x^{p}} \}$ (2)
Observing (2) we it is evident that integral (1) will converge only if $\displaystyle p>1$, and the same is for the series $\displaystyle \sum_{n=1}^{\infty} \frac{\ln n}{n^{p}}$...
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$