> [!NOTE] Proposition > Suppose $\sum_{n=0}^{\infty} a_{n} = A$ and $\sum_{n=0}^{\infty} b_{n} = B$ are [[Absolutely Convergent Series|absolutely convergent series]]. Then $AB= \sum_{n=0}^{\infty} \left( \sum_{k=0}^{n} a_{k}b_{n-k} \right)$ STS $\sum_{n=0}^{2m} \left( \sum_{k=0}^{n} a_{k} b_{n-k} \right) - \left( \sum_{n=0}^{m} a_{n} \right) \left( \sum_{n=0}^{m} b_{n} \right) \to 0 $as $m\to \infty$. We have $\begin{align} \sum_{n=0}^{2m} \left( \sum_{k=0}^{n} a_{k} b_{n-k} \right) - \left( \sum_{n=0}^{m} a_{n} \right) \left( \sum_{n=0}^{m} b_{n} \right) &= \sum_{i+j \leq 2m} a_{i} b_{j} - \sum_{i\leq m,j\leq m} a_{i} b_{j} \\ &= \end{align} $