In other words, it is the sequence whose associated formal power series is the product of the two series similarly associated to and .
A particularly important example is to consider the sequences to be terms of two strictly formal (not necessarily convergent) series
for n = 0, 1, 2, ...
"Formal" means we are manipulating series in disregard of any questions of convergence. These need not be convergent series. See in particular formal power series.
One hopes, by analogy with finite sums, that in cases in which the two series do actually converge, the sum of the infinite series
is equal to the product
just as would work when each of the two sums being multiplied has only finitely many terms. This is not true in general, but see Mertens' Theorem and Cesàro's theorem below for some special cases.
Convergence and Mertens' theorem
Let and be real sequences. It was proved by Franz Mertens that if the series converges to B and the series converges absolutely to A then their Cauchy product converges to AB. It is not sufficient for both series to be conditionally convergent. For example, the sequences are conditionally convergent but their Cauchy product does not converge.
Proof of Mertens' theorem
Let , and denote the partial sums
by rearrangement. So
Fix ε > 0. Since is absolutely convergent and is convergent then there exists an integer N such that for all we have
and an integer M such that for all it holds that
(since the series converges, the sequence must converge to 0). Also, there exists an integer L such that if then
Therefore for we have
By the definition of convergence of a series as required.
Suppose for all i > n and for all . Here the Cauchy product of and is readily verified to be . Therefore, for finite series (which are finite sums), Cauchy multiplication is direct multiplication of those series.
- For some , let and . Then
by definition and the binomial formula. Since, formally, and , we have shown that . Since the limit of the Cauchy product of two absolutely convergent series is equal to the product of the limits of those series, we have proven the formula for all .
- As a second example, let for all . Then for all so the Cauchy product does not converge.
In cases where the two sequences are convergent but not absolutely convergent, the Cauchy product is still Cesàro summable. Specifically:
If , are real sequences with and then
This can be generalised to the case where the two sequences are not convergent but just Cesàro summable:
For and , suppose the sequence is summable with sum A and is summable with sum B. Then their Cauchy product is summable with sum AB.
All of the foregoing applies to sequences in (complex numbers). The Cauchy product can be defined for series in the spaces (Euclidean spaces) where multiplication is the inner product. In this case, we have the result that if two series converge absolutely then their Cauchy product converges absolutely to the inner product of the limits.
Relation to convolution of functions
One can also define the Cauchy product of doubly infinite sequences, thought of as functions on . In this case the Cauchy product is not always defined: for instance, the Cauchy product of the constant sequence 1 with itself, is not defined. This doesn't arise for singly infinite sequences, as these have only finite sums.
One has some pairings, for instance the product of a finite sequence with any sequence, and the product . This is related to duality of Lp spaces.
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