# Find the value of p for the convergent series.

• March 15th 2010, 07:28 PM
racewithferrari
Find the value of p for the convergent series.
$

\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.
• March 15th 2010, 08:06 PM
chisigma
The sum has to be compared with the integral...

$\int_{1}^{\infty} \frac{\ln x}{x^{p}}\cdot dx$ (1)

Integrating by parts we have...

$\int_{1}^{\infty} \frac{\ln x}{x^{p}}\cdot dx = \frac{1}{-p+1}\cdot \{|x^{-p+1}\cdot \ln x|_{1}^{\infty} - \int_{1}^{\infty} \frac{dx}{x^{p}} \}$ (2)

Observing (2) we it is evident that integral (1) will converge only if $p>1$, and the same is for the series $\sum_{n=1}^{\infty} \frac{\ln n}{n^{p}}$...

Kind regards

$\chi$ $\sigma$