What values of p make the series converge?

[razored]

Homework Statement

Find the positive values of p for which the series converges.
\Sigma_{n=2}^{\infty} \frac{1}{n( \ln (n)^{p})}

Homework Equations

1/n^p converges if p>1, and diverges if =<1

The Attempt at a Solution

Don't know where to begin

 

[Dick]

Try an integral test.

 

[razored]

Doesn't the integral tell me only whether it converges or diverges? I need to actually determine the values of p for which that will converge.

 

[Dick]

Whether the integral converges or diverges will depend on the value of p.

 

[razored]

Is it two? Are we also assuming that p will be a positive integer rather than any positive real number?

 

[Dick]

What do you get if you integrate 1/(x*(ln(x))^p)? Sure, take p to be positive. If it's not then the integral definitely diverges.

 

[razored]

http://texify.com/img/%5CLARGE%5C%21%5Cint_2%5E%7B%5Cinfty%7D%20%5Cfrac%7Bdx%7D%7Bx%20%5Cln%28x%29%5E%7Bp%7D%7D%20%3D%20%20%20%5Cleft%5B%20%5Cfrac%7B%20%5Cln%20%28%5Cln%28x%29%5E%7Bp%2B1%7D%29%7D%7B%28p%2B1%29%7D%20%5Cright%5D_2%5E%7B%5Cinfty%7D.gif
What do I do now?

 

[e(ho0n3]

Have you considered working with a Taylor approximation of ln(1 + n).

 

[razored]

Have you considered working with a Taylor approximation of ln(1 + n).

That is taught later in the chapter.

 

[e(ho0n3]

http://texify.com/img/%5CLARGE%5C%21%5Cint_2%5E%7B%5Cinfty%7D%20%5Cfrac%7Bdx%7D%7Bx%20%5Cln%28x%29%5E%7Bp%7D%7D%20%3D%20%20%20%5Cleft%5B%20%5Cfrac%7B%20%5Cln%20%28%5Cln%28x%29%5E%7Bp%2B1%7D%29%7D%7B%28p%2B1%29%7D%20%5Cright%5D_2%5E%7B%5Cinfty%7D.gif
What do I do now?

What happens to ln(ln(x)^{p+1}) as x --> infty?

That is taught later in the chapter.

OK, nevermind.

 

[Dick]

http://texify.com/img/%5CLARGE%5C%21%5Cint_2%5E%7B%5Cinfty%7D%20%5Cfrac%7Bdx%7D%7Bx%20%5Cln%28x%29%5E%7Bp%7D%7D%20%3D%20%20%20%5Cleft%5B%20%5Cfrac%7B%20%5Cln%20%28%5Cln%28x%29%5E%7Bp%2B1%7D%29%7D%7B%28p%2B1%29%7D%20%5Cright%5D_2%5E%7B%5Cinfty%7D.gif
What do I do now?

That's not right. Substitute u=ln(x).