Principles of Mathematical Analysis, Rudin, 3th ed, Chapter 3, Problem 13

Problem

Prove that the Cauchy product of two absolutely convergent series converges absolutely.

Answer

Let $\sum a_n, \sum b_n$ be two absolutely convergent series. There Cauchy product is defined as $c_n:=\sum_{k=0}^n a_kb_{n-k}$.

$$ \sum_{n=0}^N |c_n| \leq \sum_{n=0}^N \Bigg|\sum_{k=0}^n a_kb_{n-k}\Bigg| \leq \sum_{n=0}^N \sum_{k=0}^n |a_k||b_{n-k}| = \sum_{n=0}^N |a_n|\sum_{k=0}^{N-n} |b_k| \leq \sum_{n=0}^\infty |a_n| \sum_{k=0}^\infty |b_k| < \infty.$$

Since $C_N:= \sum_{n=0}^N |c_n|$ is bounded and increasing, $C_N$ convergence, and therefore, $\sum c_n$ converges absolutely.


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