Editing Series (mathematics) (section)

== Operations ==

=== Series addition ===
The addition of two series <math display=inline>a_0 + a_1 + a_2 + \cdots </math> and <math display=inline>b_0 + b_1 + b_2 + \cdots </math> is given by the termwise sum<ref name=":422" /><ref name=":242">{{harvnb|Apostol|1967|pp=385–386}}</ref><ref name=":9">{{Cite book |last1=Saff |first1=E. B. |title=Fundamentals of Complex Analysis |last2=Snider |first2=Arthur D. |publisher=Pearson Education |year=2003 |isbn=0-13-907874-6 |edition=3rd |pages=247–249}}</ref><ref>{{harvnb|Rudin|1976|p=72}}</ref> <math display=inline>(a_0 + b_0) + (a_1 + b_1) + (a_2 + b_2) + \cdots \,</math>, or, in summation notation,
<math display=block>\sum_{k=0}^{\infty} a_k + \sum_{k=0}^{\infty} b_k  = \sum_{k=0}^{\infty} a_k + b_k.  </math> 

Using the symbols <math>s_{a, n}  </math> and <math>s_{b, n}  </math> for the partial sums of the added series and <math>s_{a + b, n}  </math> for the partial sums of the resulting series, this definition implies the partial sums of the resulting series follow <math>s_{a + b, n} = s_{a, n} + s_{b, n}.</math> Then the sum of the resulting series, i.e., the limit of the sequence of partial sums of the resulting series, satisfies
<math display=block>\lim_{n \rightarrow \infty} s_{a + b, n} = \lim_{n \rightarrow \infty} (s_{a, n} + s_{b, n}) = \lim_{n \rightarrow \infty} s_{a, n} + \lim_{n \rightarrow \infty} s_{b , n},</math>
when the limits exist. Therefore, first, the series resulting from addition is summable if the series added were summable, and, second, the sum of the resulting series is the addition of the sums of the added series. The addition of two divergent series may yield a convergent series: for instance, the addition of a divergent series with a series of its terms times <math>-1</math> will yield a series of all zeros that converges to zero. However, for any two series where one converges and the other diverges, the result of their addition diverges.<ref name=":242" />

For series of real numbers or complex numbers, series addition is [[Associative property|associative]], [[Commutative property|commutative]], and [[invertible]]. Therefore series addition gives the sets of convergent series of real numbers or complex numbers the structure of an [[abelian group]] and also gives the sets of all series of real numbers or complex numbers (regardless of convergence properties) the structure of an abelian group.

=== Scalar multiplication ===
The product of a series <math display=inline>a_0 + a_1 + a_2 + \cdots </math> with a constant number <math>c</math>, called a [[Scalar (mathematics)|scalar]] in this context, is given by the termwise product<ref name=":242" /> <math display=inline>ca_0 + ca_1 + ca_2 + \cdots </math>, or, in summation notation,

<math display=block>c\sum_{k=0}^{\infty} a_k = \sum_{k=0}^{\infty} ca_k.  </math>

Using the symbols <math>s_{a, n}  </math> for the partial sums of the original series and <math>s_{ca, n}  </math> for the partial sums of the series after multiplication by <math>c</math>, this definition implies that <math>s_{ca, n} = c s_{a, n}  </math> for all <math>n,  </math> and therefore also <math display=inline>\lim_{n \rightarrow \infty} s_{ca, n} = c \lim_{n \rightarrow \infty} s_{a, n},  </math>when the limits exist. Therefore if a series is summable, any nonzero scalar multiple of the series is also summable and vice versa: if a series is divergent, then any nonzero scalar multiple of it is also divergent.

Scalar multiplication of real numbers and complex numbers is associative, commutative, invertible, and it [[Distributive property|distributes over]] series addition.  

In summary, series addition and scalar multiplication gives the set of convergent series and the set of series of real numbers the structure of a [[real vector space]]. Similarly, one gets [[complex vector space]]s for series and convergent series of complex numbers. All these vector spaces are infinite dimensional.

=== 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]].