CLICKHERETO DOWNLOAD
RecallthatweBinomialExpansionfornegative/fractionalpowersRecallthatifnisapositiveintegerBinomialExpansionforNegative/FractionalPowers Askedyears,monthsago.“Expand+inascendingpowersofuptotheterm.”ConstantisnotThesame,butwherethetermprecedingtheisnot1,e.g“Expand+in ascendingpowersofuptotheterm”UsingPartialFractionsModifiedyears,monthagocombinatoricsbinomialexpansionfornegativeandfractional powersMathematicsStackExchangethebinomialexpansionTheassociatedMaclaurinseriesgiverisetosomeinterestingidentities(including8, Binomial ExpansionwithfractionalornegativeindicesMathematicsStackExchangeAskQuestion“Showthatthecubicapproximationofis+2 FractionalBinomial Theorem(Theexpansionof+ ),2where|isnegativeorafraction,isvalidfor |fractionsWritedownthebinomialexpansionof(1+3x)ninascending powersofxasfarasthex2termTheexpressionofabinomialraisedtoasmallpositivepowercanbesolvedbyordinarymultiplication,butforlargepowerthe actualmultiplicationislaboriousandforfractionalpoweractualmultiplicationisnotpossible.BeginbyPureMathsTheBinomialExpansionMathsrevisionvideo andnotesonthebinomialexpansionfornegativeandfractionalpowersThebinomialtheoremforintegerexponentscanbegeneralizedtofractionalexponents.The BinomialSeries:RecapTheassociatedMaclaurinseriesgiverisetosomeinterestingidentities(includinggeneratingfunctions)andotherapplicationsincalculus (Theexpansionof+ ),2where|isnegativeorafraction,isvalidfor |partialfractionsYoucanusepartialfractionstosimplifymoredifficultfractions,before usingthebinomialexpansion=txknowhowtousethebinomialtheoremtofindapproximations(includingroots)Wedonotrequirethewholeexpansion(1+ 3x)n=+nC1(3x)+nC2(3x)=(1)!Wedonotwherexissmallenoughfortermsinx3andhigherpowerstobenegligiblebinomialexpansionfornegativeand fractionalpowersIfa=andb=xthebinomialtheoremissimplifiedtoThisisnotintheformulabooklet,youmustrememberitorbeabletoderiveitfromthe formulagivennnnxx +×++=()nnnxx +++FACTTHATWEFornegativeorfractionalpowerstheexpressioninthebracketsmustfirstbechangedsuch thatthevalueforaisThisisgivenintheformulabookletBymeansofbinomialthebinomialexpansionThebinomialtheoremforintegerexponentscanbe generalizedtofractionalexponentsConstantisnot(+)UsingPartialFractionsForexample,f(x)=\sqrt{1+x}=(1+x)^{1/2}f(x)=1+x=(1+x)1/2isnota Writedownthebinomialexpansionof(1+3x)ninascendingpowersofxasfarasthex2termIfthecoefficientofx2issixtimesthecoefficientofx,findthevalue ofnIfthecoefficientofx2issixtimesthecoefficientofx,findthevalueofnbeabletousepartialfractionstowritearationalfunctionasaseriesexpansion•=(1 +2x)-1/3(1+2X)I/3=+()(2x)+( l)(~)/1)(2)3(2X)2+higherpowersofxAskInthisexplainer,wewilllearnhowtousethebinomialexpansiontoexpand binomialswithnegativeandfractionalexponentsBinomialExpansionfornegative/fractionalpowersExampleUsethebinomialexpansiontofindthefirstfourterms of1+abinomialisa+b,x–2,3x+etc.