Cauchy Product

DEFINITION

Notation

In the definition we will use the following notation. If

z:\mathbb{N}	o \mathbb{R}

is a Sequence , the series derived from

z

\sum z

, is a sequence whose ''n''th term is

\sum_{i=0}^n z_i

(see Series (mathematics)#Formal Definition ). If

\sum z	o Z

so that the Limit of the series exists, we speak interchangeably of the sequence and its limit, the context making clear which we mean.

Definition

Given sequences

x,y: \mathbb{N}	o \mathbb{R}

define a sequence

C(x,y)

by, for all

n\in\mathbb{N}

C(x,y)(n) = \sum_{i=0}^n x_i y_{n-i}

The series derived from the sequence

C(x,y)

\sum C(x,y)

, is the Caucy product of the series

\sum x

and

\sum y

.

EXAMPLES

Finite series

x_i = 0

for all

i>n

and

y_i = 0

for all

i>m

. Here the Cauchy product of

\sum x

and

\sum y

is readily verified to be

(x_0+\dots + x_n)(y_0+\dots+y_m)

. Therefore, for finite series (which are finite sums), Cauchy multiplication is direct multiplication of those series.

Infinite Series

For some $a,b\in\mathbb{R}$ , let $x_n = rac{a^n}{n!}$ and $y_n = rac{b^n}{n!}$ . Then

C(x,y)(n) = \sum_{i=0}^nrac{a^i}{i!}rac{b^{n-i}}{(n-i)!} = rac{(a+b)^n}{n!}

by definition and the Binomial Formula . Since, Formally ,

\exp(a) = \sum x

and

\exp(b) = \sum y

, we have shown that

\exp(a+b) = \sum C(x,y)

. Since the limit of the Cauchy product of two Absolutely Convergent series is equal to the product of the limits of those series (see Below ), we have proven the formula

\exp(a+b) = \exp(a)\exp(b)

for all

a,b\in\mathbb{R}

As a second example, let $x(n) = 1$ for all $n\in\mathbb{N}$ . Then $C(x,x)(n) = n+1$ for all $n\in\mathbb{N}$ so the Cauchy product $\sum C(x,x) = (1,1+2,1+2+3,1+2+3+4,\dots)$ and it does not converge.

CONVERGENCE AND MERTENS' THEOREM

Let ''x,y'' be real sequences. It was proved by Franz Mertens that if the series

\sum y

Converges to ''Y'' and the series

\sum x

Converges Absolutely to ''X'' then their Cauchy product

\sum C(x,y)

converges to ''XY''. It is not sufficient for both series to be Conditionally Convergent . For example, the sequence

x_n = (-1)^n /n

generates a conditionally convergent series but the sequence

C(x,x)

does not converge to 0. Here is a proof.

Proof of Mertens' Theorem

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Cauchy Product