The Cauchy product of two series: $c_n = \sum_{k=0}^{n} a_k b_{n-k}$

Let $(a_k)$ and $(b_k)$ be sequences of reals (Series, partial sums, convergence and the sum, divergence, and the tail series). The Cauchy product of $\sum a_k$ and $\sum b_k$ is the series $\sum c_n$ of the sequence

$c_n \;:=\; \sum_{k=0}^{n} a_k\, b_{n-k} \qquad (n \in \mathbb{N}),$

a finite sum of $n+1$ terms in the sense of Finite sums and finite products, by recursion. Each index $n - k$ occurring here is a natural number, because $k$ runs over $0, \dots, n$; and $c_0 = a_0 b_0$.

The definition uses only the two sequences of terms. No convergence is assumed and none is asserted: $\sum c_n$ is a series formed from $(a_k)$ and $(b_k)$, and whether it converges, and to what, is the subject of Mertens' theorem: if $\sum a_k$ converges absolutely to $A$ and $\sum b_k$ converges to $B$, their Cauchy product converges to $AB$ and If $\sum a_k$ and $\sum b_k$ both converge absolutely then their Cauchy product converges absolutely, with sum $AB$, while FALSE: the Cauchy product of two convergent series converges shows that convergence of both factors is not enough.

Why these coefficients. Reading $\sum a_k x^k$ and $\sum b_k x^k$ as formal power series and multiplying them term by term, the coefficient of $x^n$ collects exactly the products $a_k b_{n-k}$ with $k + (n-k) = n$. So $c_n$ is the coefficient one is forced to write down if the product of two series is to behave like the product of two polynomials, and the results on this page say when that formal operation computes the product of the two sums.

• Only the two sequences of terms enter. The construction is a rule on sequences, and every result below is stated for the sequence $(c_n)$ it produces. The Cauchy product of $\sum b_k$ with $\sum a_k$ is formed by the same rule with the roles exchanged, giving $\sum_{k=0}^{n} b_k a_{n-k}$; that this is the same number as $c_n$ is the reversal invariance of a finite sum, which is not among the laws of Laws of finite sums and finite products and is not used anywhere on this page. Each statement below therefore says which factor carries which hypothesis, rather than appealing to symmetry.

• The definition is stated for series indexed from $0$, as every series on this page is (Series, partial sums, convergence and the sum, divergence, and the tail series). For families from a general starting index the Cauchy product is formed after shifting both families to $\mathbb{N}$, as Series, partial sums, convergence and the sum, divergence, and the tail series prescribes; the shift changes which products appear in $c_n$, so the starting indices have to be said, and they are said wherever this construction is used below.

• Nothing here is a product of sums. The symbol $\sum c_n$ names a new series built from the terms, not the number $\bigl(\sum_{k=0}^{\infty}a_k\bigr)\bigl(\sum_{k=0}^{\infty}b_k\bigr)$, which may not even be defined. Identifying the two is a theorem with hypotheses.