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PDF-6SeriesWecallanormedspace(X;kk)aBanachspaceprovidedthateveryCauchyseq

PDF-6SeriesWecallanormedspace(X;kk)aBanachspaceprovidedthateveryCauchyseq

Author : jane-oiler | Published Date : 2016-03-09

kk11nXk21 k11 k111 n2SothesequencesnofpartialsumsisboundedinRandsinceitisalsoincreasingitconvergesExample65TheharmonicseriesP1kdivergesIndeedifsnPnk11kthenjs2nsnj1 n1

k(k1)=1+nXk=21 k11 k=1+11 n2:So thesequence(sn)ofpartialsumsisboundedinRandsinceitisalsoincreasing itconverges.Example6.5.TheharmonicseriesP1=kdiverges.Indeed ifsn=Pnk=11=k thenjs2nsnj=1 n+1+::

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6SeriesWecallanormedspace(X;kk)aBanachspaceprovidedthateveryCauchyseq: Transcript

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