Summation methods and entropy

Question

I am aware of the theory of divergent series, but don't know much of it. If you have a text to recommend, I'd be glad to hear it. Suppose I have an infinite-dimensional probability vector $\mathbf{p} = (p_{n})_{n \in \mathbb{N}} \in [0, 1)^{\mathbb{N}}$, where $\sum_{n = 1}^{\infty} p_{n} = 1$. Then is there a convenient summation method (by "summation method," I mean something like Cesaro; I apologize if that's not the canonical term) for which $\sum_{n = 1}^{\infty} p_{n} \log p_{n}$ would converge for a larger class of $ \mathbf{p}$?