I came across a question about convergence tests and I'm not sure how to show that
$\sum_{n=2}^{\infty}\frac{1}{n \log(n)^p}$
converges for $p>1$. I found that the integral test works but I haven't learnt it yet. I have learnt the comparison test, so I tried that but couldn't find any series that converges to which I can compare.
So my question is both how can I show convergence here with comparison test and also: Is there some list of convergent series that I can compare to?
PS: I also found out that this is called a ln series
Also, this is not a duplicate of this question. (The power is on the $\log$ here)
Best Answer
HINT Use the Cauchy Condensation Test: $\sum a_n$ converge iff $\sum 2^na_{2n}$ converge.
So $\sum \frac{1}{n \log(n)^p}$ converge iff $\sum 2^n\frac{1}{2^n \log(2^n)^p}$ converge.
While $\sum 2^n\frac{1}{2^n \log(2^n)^p}=\sum \frac{1}{(n\log 2)^p}=\frac{1}{(\log 2)^p}\sum\frac{1}{n^p}$