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I am reading chapter 3 in Rudin's Principles of Mathematical Analysis. In it, Rudin defines the Cauchy product of two series. That is, given $\sum a_n$ and $\sum b_n$, $c_n=\sum_{k=0}^n a_k b_{n-k}$ and $\sum c_n$ is defined as the product of the series.

What I don't understand is the motivation for the particular product. You could define the product as follows: Say $A_n=\sum_{k=0}^na_n$ and $B_n=\sum_{k=0}^nb_n$. So that $A_n B_n=\sum_{k=0}^n\sum_{i=0}^n a_k b_i=C_n$

Further assume that $A_n \rightarrow A$ and $B_n \rightarrow B$, We'll have that $A_n B_n \rightarrow AB$ since the multiplication of two convergent series is convergent.

What am I missing here?

Bernard
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daniel
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    Possibly because that's how you define the product of two power series. – Noah Olander Jul 29 '15 at 13:08
  • There's a difference between $\sum_{k=0}^n\sum_{j=0}^k$ and $\sum_{k=0}^n\sum_{j=0}^n$ – Noam Shalev - nospoon Jul 29 '15 at 13:15
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    In your second paragraph the multiplicands $\sum a_n$ and $\sum b_n$ are completely independent, e.g. you can write $C_n=\left(\sum a_k\right)\left(\sum b_k\right)$. That's trivial. The Cauchy product combines terms in a non-trivial way. Cauchy products can be useful. – pshmath0 Jul 29 '15 at 13:16
  • Thanks for the comments! I do see that the cauchy sequence combines terms in a non-trivial way.. – daniel Jul 29 '15 at 13:19

2 Answers2

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You can easily see that $$\begin{matrix}(a_0 +a_1 +a_2 + \dots )(b_0 + b_1 + b_2 +\dots ) = &a_0b_0 &+ \\ & a_0b_1 + a_1b_0 &+ \\ & a_0b_2 + a_1b_1 + a_2b_0 & + \\ & \dots \end{matrix}$$

As you can see, this rule can be written in a compact way as $$\left( \sum_{i \ge 0} a_i \right) \left( \sum_{j \ge 0} b_j \right) = \sum_{n \ge 0} \sum_{i =0}^n a_ib_{n-i}$$

Crostul
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Suppose you have two power series $\,\sum a_n x^n$ and $\,\sum b_nx^n$. Then the Cauchy product gives the coefficient of the term of degree $n$. It is a generalisation of the product of two polynomials, when these polynomials are ordered by degree.

Furthermore, you did not define a series, but a sequence.

Bernard
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