CSE235 Introduction Sequences Summations Series Sequences De nition AsequenceisafunctionfromasubsetofintegerstoasetSWeusethenotationsfangfang1nfang1n0fang1n0Eachaniscalledthenthtermofthesequenc ID: 185248
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Sequences&Summations CSE235 Introduction Sequences Summations Series Sequences Denition AsequenceisafunctionfromasubsetofintegerstoasetS.Weusethenotation(s):fangfang1nfang1n=0fang1n=0Eachaniscalledthen-thtermofthesequence. Werelyoncontexttodistinguishbetweenasequenceandaset;thoughthereisaconnection. 3/15 Sequences&Summations CSE235 Introduction Sequences Summations Series Summations II Sometimeswecanexpressasummationinclosedform.Geometricseries,forexample: Theorem Fora;r2R;r6=0,nXi=0ari=(arn+1�a r�1ifr6=1(n+1)aifr=1 11/15 Sequences&Summations CSE235 Introduction Sequences Summations Series Summations III Doublesummationsoftenarisewhenanalyzinganalgorithm.nXi=1iXj=1aj=a1+a1+a2+a1+a2+a3+a1+a2+a3++anSummationscanalsobeindexedoverelementsinaset.Xs2Sf(s)Table2onPage232(Rosen)hasusefulsummations. 12/15 Sequences&Summations CSE235 Introduction Sequences Summations Series Series Whenwetakethesumofasequence,wegetaseries.We'vealreadyseenaclosedformforgeometricseries.Someotherusefulclosedformsincludethefollowing.uXi=l1=u�l+1;forlunXi=0i=n(n+1) 2nXi=0i2=n(n+1)(2n+1) 6nXi=0ik1 k+1nk+1nXi=0 i= n+1�1 �1; 6=1nXi=1loginlogn 13/15