# Convergence of Series

#### razored

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

2. Homework Equations

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

3. The Attempt at a Solution
Don't know where to begin

Related Calculus and Beyond Homework Help News on Phys.org

#### Dick

Homework Helper
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

Homework Helper
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

Homework Helper
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 [Broken]
What do I do now?

Last edited by a moderator:

#### 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 [Broken]
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.

Last edited by a moderator:

#### Dick

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

Last edited by a moderator:

### Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving