Editing Absolute convergence (section)

==Background==
When adding a finite number of terms, [[addition]] is both [[Associative property|associative]] and [[Commutative property|commutative]], meaning that grouping and rearrangment do not alter the final sum. For instance, <math>(1+2)+3</math> is equal to both <math>1+(2+3)</math> and <math>(3+2)+1</math>. However, associativity and commutativity do not necessarily hold for infinite sums. One example is the [[alternating harmonic series]]

<math>S = \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n}=1-\frac12+\frac13-\frac14+\frac15-\frac16+\cdots </math>

whose terms are fractions that alternate in sign. This series is [[Convergent series|convergent]] and can be evaluated using the [[Maclaurin series]] for the function <math>\ln(1+x) </math>, which converges for all <math>x </math> satisfying <math>-1<x\leq1 </math>:

<math>\ln(1+x)=\sum_{n=1}^\infty \frac{(-1)^{n+1} x^n}{n}=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\cdots </math>

Substituting <math>x=1 </math> reveals that the original sum is equal to <math>\ln2 </math>. The sum can also be rearranged as follows:

<math>S=\left(1-\frac12\right)-\frac14+\left(\frac13-\frac{1}{6}\right)-\frac{1}{8}+\left(\frac15-\frac{1}{10}\right)-\frac{1}{12}+\cdots </math>

In this rearrangement, the [[Multiplicative inverse|reciprocal]] of each [[Parity (mathematics)|odd number]] is grouped with the reciprocal of twice its value, while the reciprocals of every multiple of 4 are evaluated separately. However, evaluating the terms inside the parentheses yields 

<math>S=\frac12-\frac14+\frac16-\frac18+\frac{1}{10}-\frac{1}{12}+\cdots </math> 

or half the original series. The violation of the associativity and commutativity of addition reveals that the alternating harmonic series is [[Conditional convergence|conditionally convergent]]. Indeed, the sum of the absolute values of each term is <math display="inline">1+\frac12+\frac13+\frac14+\cdots </math>, or the divergent [[Harmonic series (mathematics)|harmonic series]]. According to the [[Riemann series theorem]], any conditionally convergent series can be permuted so that its sum is any finite real number or so that it diverges. When an absolutely convergent series is rearranged, its sum is always preserved.