Editing Series (mathematics) (section)

=== 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.