Editing Series (mathematics) (section)

== Grouping and rearranging terms ==

===Grouping===
In ordinary [[Finite summation|finite summations]], terms of the summation can be grouped and ungrouped freely without changing the result of the summation as a consequence of the [[associativity]] of addition. <math>a_0 + a_1 + a_2 = {}</math><math>a_0 + (a_1 + a_2) = {}</math><math>(a_0 + a_1) + a_2.</math> Similarly, in a series, any finite groupings of terms of the series will not change the limit of the partial sums of the series and thus will not change the sum of the series. However, if an infinite number of groupings is performed in an infinite series, then the partial sums of the grouped series may have a different limit than the original series and different groupings may have different limits from one another; the sum of <math>a_0 + a_1 + a_2 + \cdots</math> may not equal the sum of <math>a_0 + (a_1 + a_2) + {}</math><math>(a_3 + a_4) + \cdots.</math> 

For example, [[Grandi's series]] {{tmath|1-1+1-1+ \cdots}} has a sequence of partial sums that alternates back and forth between {{tmath|1}} and {{tmath|0}} and does not converge. Grouping its elements in pairs creates the series <math>(1 - 1) + (1 - 1) + (1 - 1) + \cdots = {}</math><math>0 + 0 + 0 + \cdots,</math> which has partial sums equal to zero at every term and thus sums to zero. Grouping its elements in pairs starting after the first creates the series <math>1 + (- 1 + 1) + {}</math><math>(- 1 + 1) + \cdots = {}</math><math>1 + 0 + 0 + \cdots,</math> which has partial sums equal to one for every term and thus sums to one, a different result.

In general, grouping the terms of a series creates a new series with a sequence of partial sums that is a [[subsequence]] of the partial sums of the original series. This means that if the original series converges, so does the new series after grouping: all infinite subsequences of a convergent sequence also converge to the same limit. However, if the original series diverges, then the grouped series do not necessarily diverge, as in this example of Grandi's series above. However, divergence of a grouped series does imply the original series must be divergent, since it proves there is a subsequence of the partial sums of the original series which is not convergent, which would be impossible if it were convergent. This reasoning was applied in [[Harmonic series (mathematics)#Comparison test|Oresme's proof of the divergence of the harmonic series]],<ref name=":0">{{Cite journal |last1=Kifowit |first1=Steven J. |last2=Stamps |first2=Terra A. |year= 2006 |title=The harmonic series diverges again and again |url=https://stevekifowit.com/pubs/harmapa.pdf |journal=American Mathematical Association of Two-Year Colleges Review |volume=27 |issue=2 |pages=31–43}}</ref> and it is the basis for the general [[Cauchy condensation test]].<ref name=":14" /><ref name=":17">{{harvnb|Rudin|1976|p=61}}</ref>

=== 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.