Editing Series (mathematics) (section)

=== Rearrangement ===
In ordinary finite summations, terms of the summation can be rearranged freely without changing the result of the summation as a consequence of the [[commutativity]] of addition. <math>a_0 + a_1 + a_2 = {}</math><math>a_0 + a_2 + a_1 = {}</math><math>a_2 + a_1 + a_0.</math> Similarly, in a series, any finite rearrangements of terms of a series does not change the limit of the partial sums of the series and thus does not change the sum of the series: for any finite rearrangement, there will be some term after which the rearrangement did not affect any further terms: any effects of rearrangement can be isolated to the finite summation up to that term, and finite summations do not change under rearrangement.

However, as for grouping, an infinitary rearrangement of terms of a series can sometimes lead to a change in the limit of the partial sums of the series. Series with sequences of partial sums that converge to a value but whose terms could be rearranged to a form a series with partial sums that converge to some other value are called [[conditionally convergent]] series. Those that converge to the same value regardless of rearrangement are called [[unconditionally convergent]] series.

For series of real numbers and complex numbers, a series <math>a_0 + a_1 + a_2 + \cdots</math> is unconditionally convergent [[if and only if]] the series summing the [[Absolute value|absolute values]] of its terms, <math>|a_0| + |a_1| + |a_2| + \cdots, </math> is also convergent, a property called [[absolute convergence]]. Otherwise, any series of real numbers or complex numbers that converges but does not converge absolutely is conditionally convergent. Any conditionally convergent sum of real numbers can be rearranged to yield any other real number as a limit, or to diverge. These claims are the content of the [[Riemann series theorem]].<ref name=":46">{{harvnb|Spivak|2008|pp=483–486}}</ref><ref name=":25">{{harvnb|Apostol|1967|pp=412–414}}</ref><ref>{{harvnb|Rudin|1976|p=76}}</ref>

A historically important example of conditional convergence is the [[alternating harmonic series]],

<math display=block>\sum\limits_{n=1}^\infty {(-1)^{n+1}  \over n} = 1 - {1 \over 2} + {1 \over 3} - {1 \over 4} + {1 \over 5} - \cdots,</math>
which has a sum of the [[natural logarithm of 2]], while the sum of the absolute values of the terms is the [[Harmonic series (mathematics)|harmonic series]],
<math display=block>\sum\limits_{n=1}^\infty {1 \over n} = 1 + {1 \over 2} + {1 \over 3} + {1 \over 4} + {1 \over 5} + \cdots,</math>
which diverges per the divergence of the harmonic series,<ref name=":0" /> so the alternating harmonic series is conditionally convergent. For instance, rearranging the terms of the alternating harmonic series so that each positive term of the original series is followed by two negative terms of the original series rather than just one yields<ref name=":4222">{{harvnb|Spivak|2008|p=482}}</ref> 
<math display=block>\begin{align}
&1 - \frac12 - \frac14 + \frac13 - \frac16 - \frac18
  + \frac15 - \frac1{10} - \frac1{12} + \cdots \\[3mu]
&\quad = \left(1 - \frac12\right) - \frac14
  + \left(\frac13 - \frac16\right) - \frac18
  + \left(\frac15 - \frac1{10}\right) - \frac1{12} + \cdots \\[3mu]
&\quad = \frac12 - \frac14 + \frac16 - \frac18
  + \frac1{10} - \frac1{12} + \cdots \\[3mu]
&\quad = \frac12 \left(1 - \frac12 + \frac13 - \frac14
  + \frac15 - \frac16 + \cdots \right) ,
\end{align}</math>
which is <math>\tfrac12</math> times the original series, so it would have a sum of half of the natural logarithm of 2. By the Riemann series theorem, rearrangements of the alternating harmonic series to yield any other real number are also possible.