请输入您要查询的字词:

 

单词 SumOfSeriesDependsOnOrder
释义

sum of series depends on order


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

1-12+13-14+15-16+17-18+19-110+111-112+-

is convergentMathworldPlanetmathPlanetmath and has a positive sum (=ln2; see the natural logarithmMathworldPlanetmathPlanetmath (http://planetmath.org/NaturalLogarithm2)).  Denote it by S.  We can by 12 getting the two seriesS=(1-12)+(13-14)+(15-16)+(17-18)+(19-110)+,

12S=12-14+16-18+110-+.

Then we add these two series termwise getting the sum

112S=1+13-24+15+17-28+19+111-212+.

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.

Titlesum of series depends on order
Canonical nameSumOfSeriesDependsOnOrder
Date of creation2013-03-22 14:50:59
Last modified on2013-03-22 14:50:59
Ownerpahio (2872)
Last modified bypahio (2872)
Numerical id16
Authorpahio (2872)
Entry typeExample
Classificationmsc 26A06
Classificationmsc 40A05
Related topicAbsoluteConvergence
Related topicOrderOfFactorsInInfiniteProduct
Related topicAlternatingHarmonicSeries
Related topicConditionallyConvergentSeries
Related topicConvergingAlternatingSeriesNotSatisfyingAllLeibnizConditions
Related topicFiniteChangesInConvergentSeries
Related topicFiniteChangesInConvergentSeries2
随便看

 

数学辞典收录了18232条数学词条,基本涵盖了常用数学知识及数学英语单词词组的翻译及用法,是数学学习的有利工具。

 

Copyright © 2000-2023 Newdu.com.com All Rights Reserved
更新时间:2025/5/4 17:08:35