$\displaystyle

\sum\limits_{n = 1}^\infty {ln (n) / n^{ p} }

$

I am using integral test asgoes to infinity. I am not familiar with finding the pt

Any help.

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- Mar 15th 2010, 07:28 PMracewithferrariFind the value of p for the convergent series.
$\displaystyle

\sum\limits_{n = 1}^\infty {ln (n) / n^{ p} }

$

I am using integral test asgoes to infinity. I am not familiar with finding the p*t*

Any help. - Mar 15th 2010, 08:06 PMchisigma
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

$\displaystyle \chi$ $\displaystyle \sigma$