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Determine all values of $p$ and $q$ for which the following series converges: $$\sum_{k=2}^{\infty} \frac{1}{k^q (\ln k)^p}$$ Hints : Consider the three case $q>1$, $q=1$, $q<1$.

I understand how to find them when $q=1$. I am struggling with $q<1$ and $q>1$, hope anyone can help.

megan
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  • I edited your question to implement LaTeX. You might get better help if you use LaTeX whenever possible and if you use grammatically correct and sensical sentences. – superAnnoyingUser Feb 13 '14 at 10:05
  • $q=1$ is the only hard case… for $q>1$ and $q<1$, the log term is irrelevant (since $\ln k$ is $o(k^{\varepsilon})$ for any $\varepsilon > 0$). So the series diverges for $q<1$ and converges for $q>1$. – mjqxxxx Aug 01 '14 at 00:03
  • hint for q=1 https://math.stackexchange.com/questions/2227199/does-the-series-sum-n-2-infty-frac1n-lnn2-converge-or-diver?noredirect=1&lq=1 – Sine Wu Jan 11 '19 at 10:15

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In these two cases you can compare the given series with $\sum_{k=2}^\infty\frac{1}{k^q}$ if $q>1$ and with $\sum_{k=2}^\infty\frac{1}{k^{q+\frac{1-q}{2}}}$ if $q<1$

kmitov
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