sum of series depends on order

According to the Leibniz’ test (, the alternating seriesMathworldPlanetmath


is convergentMathworldPlanetmathPlanetmath and has a positive sum (=ln2; see the natural logarithmMathworldPlanetmathPlanetmath (  Denote it by S.  We can by 12 getting the two series S=(1-12)+(13-14)+(15-16)+(17-18)+(19-110)+,


Then we add these two series termwise getting the sum


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 valuesPlanetmathPlanetmathPlanetmath of its is the divergent harmonic seriesMathworldPlanetmath.

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