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9.6AlternatingSeries;ConditionalConvergence
AlternatingSeriesSerieswhosetermsalternatebetweenpositiveandnegative,calledalternatingseries,areofspecialimportance.Someexamplesare:
AlternatingSeriesIngeneral,analternatingserieshasoneofthefollowingtwoformswherethebksareassumedtobepositiveinbothcases.
AlternatingSeriesThefollowingtheoremisthekeyresultonconvergenceofalternatingseries.
Theevenpartialsumss2,s4,s6,...
areincreasing.Theoddpartialsumss1,s3,s5,...
aredecreasing.
ALTERNATINGSERIESTEST—PROOFFirst,weconsidertheevenpartialsums:s2=b1–b2≥0 sinceb2≤b1
s4=s2+(b3–b4)≥s2 sinceb4≤b3Ingeneral,s2n=s2n–2+(b2n–1–b2n)≥s2n–2 sinceb2n≤b2n–1Thus, 0≤s2≤s4≤s6≤…≤s2n≤…
ALTERNATINGSERIESTEST—PROOFHowever,wecanalsowrite:s2n=b1–(b2–b3)–(b4–b5)–… –(b2n–2–b2n–1)–b2nEveryterminbracketsispositive.So,s2n≤b1foralln.Thus,thesequence{s2n}ofeven
partialsumsisincreasingandboundedabove.Therefore,itisconvergentbytheMonotoneSequenceTheorem.
ALTERNATINGSERIESTEST—PROOFLet’scallitslimits,thatis,Now,wecomputethelimitoftheoddpartialsums:
Example1Usethealternatingseriestesttoshowthatthefollowingseriesconverge.(a)(b)
Example1Usethealternatingseriestesttoshowthatthefollowingseriesconverge.(a)(b)(a)Thislookslikethedivergentharmonicseries.However,thisseriesconvergessincebothconditionsinthealternatingseriestestaresatisfied.
Example1Usethealternatingseriestesttoshowthatthefollowingseriesconverge.(a)(b)(b)Thisseriesconvergessincebothconditionsinthealternatingseriestestaresatisfied.
ApproximatingSumsofAlternatingSeriesThefollowingtheoremisconcernedwithth
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