Cauchy product

Definition

Let $\sum_{n=0}^{\infty} a_n$ and $\sum_{n=0}^{\infty} b_n$ be real series. For each $n$, let $c_n = \sum_{k=0}^{n} a_k b_{n-k}$. The series $\sum_{n=0}^{\infty} c_n$ is the Cauchy product of the two series.

Remarks

The term $c_n$ collects all products $a_i b_j$ with $i + j = n$. For power series, the Cauchy product of $\sum a_n x^n$ and $\sum b_n x^n$ is $\sum c_n x^n$. This is the rule for multiplying polynomials. The Cauchy product of two convergent series need not converge. Mertens' theorem gives a sufficient condition.

Explanations

L1 human

To multiply two series, multiply every term of one by every term of the other. The Cauchy product groups these products by the sum of their indices.