Values of p 1/n[n(ln(n)^p)]

[runningeagle]

Hi.

I am looking at [attachment=1:3in2bfj5]2.gif[/attachment:3in2bfj5] [NOTE: Log is the naturallog]. I need to use the integral test to find the values of p for which the series converges

So, I get the integral to be [attachment=2:3in2bfj5]1.gif[/attachment:3in2bfj5]. So, I simplify this to[attachment=0:3in2bfj5]3.gif[/attachment:3in2bfj5] using the properties of logarthims, with exponents, and I cancel out the ln(n)s.

Now I have 1/[p(-p+1)] evaluated from 2 to infinity. Now, the only values I know that this diverges for are 0 and -1.

How do I find the interval(s) of convergence now?

 

[pka]

What you have is not quite right.
\(\displaystyle \int_2^\infty {\frac{{dx}}{{x\ln ^p (x)}}} = \lim _{b \to \infty } \int_2^b {\frac{{dx}}{{x\ln ^p (x)}}} = \lim _{b \to \infty } \left[ {\frac{{\ln ^{1-p} (b)}}{{1-p}} - \frac{{\ln ^{1-p} (2)}}{{1- p}}} \right]\)

Therefore, you must find a value of p so that
\(\displaystyle \lim _{b \to \infty } \left[ {\frac{{\ln ^{1-p} (b)}}{{1-p}}} \right]\) converges.