The convergence of the sum \(\sum_{k=10}^{\infty} \frac{1}{k^p (\ln(\ln(k)))^p}\) is determined by the value of \(p > 0\). The series converges when \(p > 1\) due to the comparison with the p-series \(\frac{1}{k^p}\), which converges under the same condition. The logarithmic term \((\ln(\ln(k)))^p\) does not affect convergence for large \(k\) since it remains finite and does not dominate the behavior of the series. Therefore, the critical threshold for convergence is established as \(p > 1\).
PREREQUISITESMathematics students, particularly those studying calculus or real analysis, educators teaching series convergence, and anyone interested in advanced mathematical series and their properties.
Jacob_ said:Homework Statement
For which p > 0 does the sum
[itex]\displaystyle\sum\limits_{k=10}^∞ \frac{1}{k^p(ln(ln(k)))^p}[/itex]
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.