Let $$f: \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}$$ be a function such that $$\sum_{(n,m) \in \mathbb{N} \times \mathbb{N}} f(n, m)$$ is absolutely convergent. Then we have
\begin{aligned} \sum_{n=0}^{\infty} \left( \sum_{m=0}^{\infty} f(n, m) \right) &= [...] \\ &= [...] \\ &= \sum_{m=0}^{\infty} \left( \sum_{n=0}^{\infty} f(n, m) \right)\end{aligned}