Editing Series (mathematics) (section)

=== 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.