Definition. (Cauchyprodukt) $\sum_{n = 0}^{\infty} c_{n}$ ist das Cauchyprodukt der Reihen $\sum_{i = 0}^{\infty} a_{i}$ und $\sum_{j = 0}^{\infty} b_{j}$ =Df

c_{n} = \sum_{i + j = n}^{} a_{i} b_{j} = \sum_{i = 0}^{n} a_{i} b_{n - i} .

Es gilt also:

$\sum c_{n} = (\sum a_{i}) \cdot (\sum b_{j})$ $= a_{0} b_{0} + (a_{0} b_{1} + a_{1} b_{0}) + (a_{0} b_{2} + a_{1} b_{1} + a_{2} b_{0}) +$ $(a_{0} b_{3} + a_{1} b_{2} + a_{2} b_{1} + a_{3} b_{0}) + \dots$ $= \sum_{n = 0}^{\infty} (\sum_{i + j = n}^{} a_{i} b_{j})$ $= \sum_{n = 0}^{\infty} (\sum_{i = 0}^{n} a_{i} b_{n - 1})$ (für $j = n - i$ ).