sum of series depends on order

According to the Leibniz’ test (http://planetmath.org/LeibnizEstimateForAlternatingSeries), the alternating series

$$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\frac{1}{6}+\frac{1}{7}-\frac{1}{8}+\frac{1}{9}-\frac{1}{10}+\frac{1}{11}-\frac{1}{12}+-\mathrm{\dots }$$

is convergent and has a positive sum ($=\ln 2$; see the natural logarithm (http://planetmath.org/NaturalLogarithm2)).  Denote it by $S$.  We can by $\frac{1}{2}$ getting the two series $S=(1-\frac{1}{2})+(\frac{1}{3}-\frac{1}{4})+(\frac{1}{5}-\frac{1}{6})+(\frac{1}{7}-\frac{1}{8})+(\frac{1}{9}-\frac{1}{10})+\mathrm{\dots },$

$\frac{1}{2}S=\frac{1}{2}-\frac{1}{4}+\frac{1}{6}-\frac{1}{8}+\frac{1}{10}-+\mathrm{\dots }.$

Then we add these two series termwise getting the sum

$1⁤\frac{1}{2}S=1+\frac{1}{3}-\frac{2}{4}+\frac{1}{5}+\frac{1}{7}-\frac{2}{8}+\frac{1}{9}+\frac{1}{11}-\frac{2}{12}+\mathrm{\dots }.$

Hence, this last series exactly the same as the original, but its sum is fifty percent greater.  This is possible because the original series is not absolutely convergent:  the series which is formed of the absolute values of its is the divergent harmonic series.

P. S.  – For justification of the used manipulations of the series, see the entry.

Title	sum of series depends on order
Canonical name	SumOfSeriesDependsOnOrder
Date of creation	2013-03-22 14:50:59
Last modified on	2013-03-22 14:50:59
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	16
Author	pahio (2872)
Entry type	Example
Classification	msc 26A06
Classification	msc 40A05
Related topic	AbsoluteConvergence
Related topic	OrderOfFactorsInInfiniteProduct
Related topic	AlternatingHarmonicSeries
Related topic	ConditionallyConvergentSeries
Related topic	ConvergingAlternatingSeriesNotSatisfyingAllLeibnizConditions
Related topic	FiniteChangesInConvergentSeries
Related topic	FiniteChangesInConvergentSeries2