Editing Series (mathematics) (section)

=== Series multiplication ===
The multiplication of two series <math>a_0 + a_1 + a_2 + \cdots </math> and <math>b_0 + b_1 + b_2 + \cdots </math> to generate a third series <math>c_0 + c_1 + c_2 + \cdots </math>, called the Cauchy product,<ref name=":7" /><ref name=":422" /><ref name=":8" /><ref name=":9" /><ref>{{harvnb|Rudin|1976|p=73}}</ref> can be written in summation notation
<math display=block>
\biggl( \sum_{k=0}^{\infty} a_k \biggr) \cdot  \biggl( \sum_{k=0}^{\infty} b_k \biggr)
= \sum_{k=0}^{\infty} c_k = \sum_{k=0}^{\infty} \sum_{j=0}^{k} a_{j} b_{k-j},
</math>
with each <math display=inline>c_k = \sum_{j=0}^{k} a_{j} b_{k-j} = {}\!</math><math>\!a_0 b_k + a_1 b_{k-1} + \cdots + a_{k-1} b_1 + a_k b_0.</math> Here, the convergence of the partial sums of the series <math>c_0 + c_1 + c_2 + \cdots </math> is not as simple to establish as for addition. However, if both series <math>a_0 + a_1 + a_2 + \cdots </math> and <math>b_0 + b_1 + b_2 + \cdots </math> are [[absolutely convergent]] series, then the series resulting from multiplying them also converges absolutely with a sum equal to the product of the two sums of the multiplied series,<ref name=":422" /><ref name=":9" /><ref>{{harvnb|Rudin|1976|p=74}}</ref>
<math display=block>\lim_{n \rightarrow \infty} s_{c, n} = \left(\, \lim_{n \rightarrow \infty} s_{a, n} \right) \cdot \left(\, \lim_{n \rightarrow \infty} s_{b , n} \right).</math>

Series multiplication of absolutely convergent series of real numbers and complex numbers is associative, commutative, and distributes over series addition. Together with series addition, series multiplication gives the sets of absolutely convergent series of real numbers or complex numbers the structure of a [[Commutative ring|commutative]] [[Ring (mathematics)|ring]], and together with scalar multiplication as well, the structure of a [[Commutative algebra (structure)|commutative algebra]]; these operations also give the sets of all series of real numbers or complex numbers the structure of an [[associative algebra]].