SeriesHandout - Math 4150 6150 Fall 2011 Review of Innite...

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Math 4150 / 6150Fall 2011Review of Infinite Series from 3100just the basics - no powers series, Taylor’s theorem, or uniform convergence1.Important infinite seriesGeometric series:n=1rnconverges⇐⇒ |r|<1. If|r|<1, thenn=1rn=r1-r.Thep-series:n=11npconverges⇐⇒p >1.[Telescoping series:If limn→∞bn=L, thenn=1(bn-bn+1) =b1-L.]2.Series TestsDefinition.Given a sequence{an}letsn=a1+· · ·+andenote itsnth partial sum, then{an}summable⇐⇒Xn=1anconverges⇐⇒{sn}converges.
Ifn=1anconverges, thenlimn→∞an= 0.2.1 Series of non-negative termsProposition 3.Ifan0andsn=a1+· · ·+an, thenXn=1anconverges⇐⇒{sn}bounded.Proposition 4(First Comparison Test).If0anbnfor all sufficiently largenN, thenXn=1bnconverges=Xn=1anconverges.Proposition 5(Second Comparison Test).Supposean>0,bn>0, andlimn→∞anbn=L6= 0, thenXn

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