For which values of p does this sum converge?

[Jacob_]

Homework Statement

For which p > 0 does the sum
\displaystyle\sum\limits_{k=10}^∞ \frac{1}{k^p(ln(ln(k)))^p}
converge?

Homework Equations

1/k^p converges for p > 1.

The Attempt at a Solution

I'm not really sure where to start. I want to use a comparison test with the p-series, but ln(ln(k)) < 1 for k < e^e, so the equation isn't greater or less than 1/k^p for the entire sum interval.

 

[sunjin09]

Homework Statement

For which p > 0 does the sum
\displaystyle\sum\limits_{k=10}^∞ \frac{1}{k^p(ln(ln(k)))^p}
converge?

Homework Equations

1/k^p converges for p > 1.

The Attempt at a Solution

I'm not really sure where to start. I want to use a comparison test with the p-series, but ln(ln(k)) < 1 for k < e^e, so the equation isn't greater or less than 1/k^p for the entire sum interval.

Convergence of the series is determined only by the asymptotic behavior of the terms in the sum, for any finite k, the term is finite, and therefore irrelevant