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?
Best Answer
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.