TITLE: Does the Shannon Entropy always exist (even for infinite distributions)?
QUESTION [5 upvotes]: Let $p : \mathbb{N} \to [0, 1]$ be a probability distribution over the naturals.
The Shannon Entropy is:
$$H = -\sum_{n=0}^\infty p(n)\log_2 p(n)$$
Does this series always converge?

I tried a little to attack this problem but it's been some time since I evaluate series. My attempt:
We known that $\sum_{n=0}^\infty p(n) = 1$, therefore it must be the case that $\lim\limits_{n\to\infty} p(n) = 0$, and since $\lim\limits_{x\to 0} x\log_2 x = 0$, for any $\varepsilon > 0$ there exists only a finite amount of terms greater than $\varepsilon$. The problem is that I can't conclude anything from this, because there are several series in which the terms go to zero but it still diverges.

REPLY [3 votes]: The answer is no. Define $p$ by
$$ p(n) = \frac{c}{(n+2)\log^2(n+2)}, $$
where $c$ is the normalizing constant. Then it follows that $-p(n)\log p(n) \sim c/n \log n$ as $n\to\infty$ and hence $H(p) = \infty$ by the limit comparison test.