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Homework Statement
Show for what real numbers p and q ## \sum \frac{1}{n^{p}ln^{q}(n)} ## diverges or converges.
Homework Equations
The Attempt at a Solution
I am kind of lost because it seems that with both subscripts p and q there are a bunch of cases you have to work through. My professor wasn't explicit in regards to what values p and q we had to consider.
My number 1 question would be: do you consider p and q covering the same set of numbers? Such as p>1 and q>1? Or do you have to also consider 0<p<1 while q>1 as well? It just seems this would take forever.
Anyways I think I have a solution for p,q > 1
## \frac{1}{n^{p}ln^{q}(n)} ## ≤ ## \frac{1}{n^{p}} ## for all n ≥ 3
since ## \frac{1}{n^{p}} ## converges (p > 1) by comparison test ## \sum \frac{1}{n^{p}ln^{q}(n)} ## converges for p,q > 1