sequences and series - Does the Shannon Entropy always exist (even for infinite distributions)?

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?

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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.