Cauchy product

The Cauchy product multiplies two sequences or infinite series by summing products of terms whose indices add to the same total. Its convergence depends on the behavior of the original series.

Convolution: An operation that combines two sequences or functions by summing products over matching indices or shifts. The Cauchy product is the discrete convolution of the two coefficient sequences.
Infinite series: A sum defined by the limit of the partial sums of an infinite sequence of terms. The Cauchy product is a rule for forming a new series from two infinite series.
Geometric series: A series whose successive terms have a constant ratio. Multiplying two convergent geometric series gives a direct example of coefficient convolution.
Hadamard product: The termwise product of two sequences or power series, pairing coefficients at identical indices. Unlike the Cauchy product, it multiplies matching coefficients instead of summing across diagonals.
Cauchy product theorem: A theorem giving conditions under which the Cauchy product of two series converges to the product of their sums. It states precisely when multiplying the original infinite sums is valid.
Absolute convergence: A series converges absolutely when the series of absolute values of its terms converges. Absolute convergence guarantees that rearranging the product's terms preserves its sum.
Sequence: An ordered list of terms indexed by integers or another specified set. The product acts first on the coefficient sequences underlying the two series.
Exponential function: The function characterized by its power series and the identity exp(x+y)=exp(x)exp(y). Its power series coefficients multiply by the Cauchy rule to recover the addition identity.
Pointwise multiplication: Multiplication of functions by multiplying their values separately at each input. For power series, pointwise multiplication leads to Cauchy convolution of coefficients within convergence domains.
Analytic function: A function locally represented by a convergent power series. Cauchy products show that sums and products of analytic functions retain local power-series representations.

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