# Potenzreihen/R/Gleiche Variable/Cauchyprodukt/Aufgabe/en

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Let ${\displaystyle {}\sum _{n=0}^{\infty }a_{n}x^{n}}$ and ${\displaystyle {}\sum _{n=0}^{\infty }b_{n}x^{n}}$ be two power series absolutely convergent in ${\displaystyle {}x\in \mathbb {R} }$. Prove that the Cauchy product of these series is exactly

${\displaystyle \sum _{n=0}^{\infty }c_{n}x^{n}{\text{ where }}c_{n}=\sum _{i=0}^{n}a_{i}b_{n-i}.}$