Show that the SERIES log k / (k^p), p>1 converges.
thanks.
Let $\displaystyle \sum{a_n}$ be an infinite series with $\displaystyle a_{n+1}\leqslant{a_n}$ and $\displaystyle a_n>0$. Then $\displaystyle \sum{a_n}$ converges iff $\displaystyle \sum{2^na_{2^n}}$ converges. So since $\displaystyle \forall{k}\in(2,\infty)~\frac{\log(k)}{k^p}>1$ and monotonically decreasing we may apply the above test. So
$\displaystyle \sum2^n\frac{\log(2^n)}{\left(2^n\right)^p}=\sum\f rac{n\log(2)}{2^{n(p-1)}}$. Now applying the Root test gives
$\displaystyle \begin{aligned}\limsup\sqrt[n]{\left|\frac{n\log(2)}{2^{n(p-1)}}\right|}&=\limsup\frac{\sqrt[n]{n\log(2)}}{2^{p-1}}\\
&=\frac{1}{2^{p-1}}\end{aligned}$
So now for a series to converge we must have that $\displaystyle \limsup\sqrt[n]{|a_n|}<1$
So solving this for p gives $\displaystyle p>1$