Prove the series is convergent for p>1:
$\sum _{k=1}^{\infty } \frac{1}{(n)(ln(n)^p)}$
The ratio test is inconclusive. I assume a comparison test is involved but am not sure how to make that happen.
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Note: I'm assuming $n=k$ in your question and $k$ must start from 2!
Use the same reduction (combine 1 term, 2 terms, 4 terms, etc.) as in the case of the p-series, and then... you get the p-series.
See Proving the convergence of the $p$-series without using the integral test? for the proof of the p-series.
$$ \sum_{n\geq2}\frac1{n \ln(n)^p}\leq\sum_{k\geq1}\frac{2^k}{2^k\ln(2^k)^{p}}=\frac{1}{\ln(2)^p}\sum_{k\geq1}\frac{1}{k^p}. $$
Technically you should say that the partial sums are all bounded above by the RHS.
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