1

Let $X\sim P$ on $A=\{2,3,\dots\}$, where $P(k)=\frac{C}{k(\log k)^2}$ for $k\geq2$ with $C$ some normalising constant. Show that $H(X)=\infty$.

My attempt so far: I have shown, by direct computation that $$H(X)=C\sum_{k=2}^\infty\frac{\log k+2\log\log k-\log C}{k(\log k)^2},$$ but I'm not sure how to show that this sum diverges. Advice would be greatly appreciated. Thanks!

verygoodbloke
  • 1,033
  • 2
    https://math.stackexchange.com/questions/574503/infinite-series-sum-n-2-infty-frac1n-log-n –  May 14 '22 at 13:23
  • 1
    The reason you started with $P(k) = \frac{c}{k(\log(k))^2}$ was that using $P(k)=\frac{c}{k(\log(k))}$ would not sum to a finite number. You wanted a PMF that decays slowly but not too slowly so that it cannot be a PMF. – Michael May 14 '22 at 14:19

0 Answers0