Convergence of Series

1. Mar 29, 2009

razored

1. The problem statement, all variables and given/known data
Find the positive values of p for which the series converges.
$$\Sigma_{n=2}^{\infty} \frac{1}{n( \ln (n)^{p})}$$

2. Relevant equations

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

3. The attempt at a solution
Don't know where to begin

2. Mar 29, 2009

Dick

Try an integral test.

3. Mar 29, 2009

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.

4. Mar 29, 2009

Dick

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

5. Mar 29, 2009

razored

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

6. Mar 29, 2009

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.

7. Mar 29, 2009

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 [Broken]
What do I do now?

Last edited by a moderator: May 4, 2017
8. Mar 29, 2009

e(ho0n3

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

9. Mar 29, 2009

razored

That is taught later in the chapter.

10. Mar 29, 2009

e(ho0n3

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

OK, nevermind.

Last edited by a moderator: May 4, 2017
11. Mar 29, 2009

Dick

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

Last edited by a moderator: May 4, 2017