Find p in R for Absolute Conv. Series: 1/(klog^pk)

[wany]

Homework Statement

find all values of p in R for which the given series converges absolutely.
$\displaystyle\sum\limits_{k=2}^{\infty} \frac{1}{klog^pk}$

Homework Equations

Ratio Test Or Root Test

The Attempt at a Solution

So I tried using the ratio test and get:
$\mathop {\lim }\limits_{k \to \infty } \frac{|a_{k+1}|}{|a_k|}=\mathop {\lim }\limits_{k \to \infty } |\frac{klog^pk}{(k+1)log^p(k+1)}|=\mathop {\lim }\limits_{k \to \infty } |\frac{log^pk}{log^p(k+1)}|$ since lim as k goes to infinity of k/(k+1) is 1.
We know that we want this limit to be less than 1.

I am stuck from this point. Any help would be appreciated (there should be absolute signs in there I am not sure why they didnt show up).

 

[Dick]

Use an integral test.

 

[wany]

ok, so i let $f(x)=\frac{1}{xlog^px}$ and then when I take the integral of this function, first I let $log^px=\frac{lnx^p}{ln10^p}$ and then I let u=lnx
so I get that the integral is $ln10^p*\frac{-u^{1-p}}{p-1}$ taken from 1 to t and taking the limit as t goes to infinity, so $ln10^p*\frac{ln(1)^{1-p}-lnt^{1-p}}{p-1}$
So we see that when p is in (1, infinity), that makes sure that $lnt^{1-p}$ goes to 0 instead of infinity and thus our series converges. Is this correct? And thank you for your help so far.

 

[Dick]

ok, so i let $f(x)=\frac{1}{xlog^px}$ and then when I take the integral of this function, first I let $log^px=\frac{lnx^p}{ln10^p}$ and then I let u=lnx
so I get that the integral is $ln10^p*\frac{-u^{1-p}}{p-1}$ taken from 1 to t and taking the limit as t goes to infinity, so $ln10^p*\frac{ln(1)^{1-p}-lnt^{1-p}}{p-1}$
So we see that when p is in (1, infinity), that makes sure that $lnt^{1-p}$ goes to 0 instead of infinity and thus our series converges. Is this correct? And thank you for your help so far.

I think that's actually pretty much it. I wouldn't worry about the logs base 10 vs ln. I think you can take log=ln if it doesn't specifically say log_10. And they did set the lower limit of the sum to n=2 for a good reason. Otherwise the first term in the sequence is undefined. But other than that it looks ok to me.

 

[wany]

Oh you it is 2 and not 1. Thank you for your help, I appreciate it.

 

[wany]

One more thing, since I want it to converge absolutely, do I need to have the integral of the absolute function?

 

[wany]

Never mind that was a stupid question, the function is positive and decreasing, so absolute does not matter in this case.