- #1

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Show that the series with general term ##a_n=\dfrac{1}{n^plog(n)^q}, n\geq 2##

converges if: ##q>0## and ##p>1##; ##q>1## and ##p=1##

diverges if: ##q>0## and ##p<1##; ##0<q\leq 1## and ##p=1##

The attempt at a solution.

I could solve the problem for the cases ##q>1## and ##p=1##, and ##0<q\leq 1## and ##p=1##:

For these two cases I can use the integral test where ##f(x)=\dfrac{1}{xlog(x)^q}##.

By making the substitution ##u=log(x)##, I got that ##\int_2^{+\infty} f(x)dx## converges if and only if q>1.

Now, I don't now how to show the convergence and divergence respectively for the other two remaining cases (q>0). I would appreciate any suggestions.