The Cauchy Product of Two Absolutely Convergent Series

In the setting of The Real Line: Standing Notation and Background for Calculus on Intervals, let partial sums, convergence and sums of series and absolute convergence be as in Series of Real Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence, and for $n\in\mathbb{N}$ let $[n]=\{1,\dots,n\}$ be the segment. Let $(a_{k})$ and $(b_{k})$ be sequences in $\mathbb{R}$ such that $\sum_{k=1}^{\infty}a_{k}$ and $\sum_{k=1}^{\infty}b_{k}$ converge absolutely; these series converge, by Series of Real Numbers with Nonnegative Terms, Comparison, Absolute Convergence and the Geometric Series (iii). For $n\in\mathbb{N}$ let

$$D_{n}=\{(i,j)\in[n]\times[n]:i+j=n+1\},\qquad c_{n}=\sum_{(i,j)\in D_{n}}a_{i}b_{j},$$

the sum over the subset $D_{n}$ of $[n]\times[n]$ (Sums and Products over a Finite Set and over an Interval (v)). Then $(c_{n})$ is a sequence in $\mathbb{R}$, by Finite Sums of Reals: Congruence, the Product Map (x,y) ↦ f(x)h(y), and the Sequence of Diagonal Sums (iii) ($a_{i}$ and $b_{j}$ are defined for $(i,j)\in D_{n}$ as $i,j\in[n]$ are natural numbers by Elements of a Segment Are Natural Numbers, and Mixed Transitivity of the Order on the Natural Numbers (i)).

(i) The series $\sum_{n=1}^{\infty}c_{n}$ converges absolutely.

(ii) The series $\sum_{n=1}^{\infty}c_{n}$ converges, and

$$\sum_{n=1}^{\infty}c_{n}=\Big(\sum_{k=1}^{\infty}a_{k}\Big)\Big(\sum_{k=1}^{\infty}b_{k}\Big).$$