For which values of p does this sum converge?


by Jacob_
Tags: converge, values
Jacob_
Jacob_ is offline
#1
Mar4-12, 11:09 PM
P: 1
1. The problem statement, all variables and given/known data
For which p > 0 does the sum
[itex]\displaystyle\sum\limits_{k=10}^∞ \frac{1}{k^p(ln(ln(k)))^p}[/itex]
converge?


2. Relevant equations
1/k^p converges for p > 1.


3. 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.
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sunjin09
sunjin09 is offline
#2
Mar5-12, 01:30 PM
P: 312
Quote Quote by Jacob_ View Post
1. The problem statement, all variables and given/known data
For which p > 0 does the sum
[itex]\displaystyle\sum\limits_{k=10}^∞ \frac{1}{k^p(ln(ln(k)))^p}[/itex]
converge?


2. Relevant equations
1/k^p converges for p > 1.


3. 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


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