Summary: Binomial Expansion...! r. where
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1 Summy: Biomil Epsio 009 M Teo ) Re-cp of Additiol Mthemtics Biomil Theoem... whee )!!(! () The fomul is ville i MF so studets do ot eed to memoise it. () The fomul pplies oly to positive itege vlues of. () Tems d c e costts d/o epessios i tems of. Specil If d e epessios i tems of, studets c use to detemie the coefficiet fo y powes of. Fo emple, i the epsio of 0 whee =, = d = 0 Studets c use to oti the geel equtio 0 0 which c e simplified to Usig this geel equtio, the coefficiet of y powes of c e esily detemied. Fo istce, if the tem idepedet of is equied, studets c detemie the vlue y settig 0- = 0. Hece, vlue is 0 d the costt c e susequetly otied usig the sme equtio. Studets should hve epeiece i this polem sice it hs ledy ee widely tested t the O levels!
2 Summy: Biomil Epsio ) Epsio of whee is o-positive itege I H Mthemtics, studets e epdig whee is o-positive itege c e egtive itege, fctio etc. The fomul i Sectio is thus ot pplicle to such epsio! Befoe studets c epd the epessio fom oe of the tems must e. Cosideig oly the tem, the iomil seies epsio is ( )!, it must fist e modified to eithe ( )...( )...!... (*) o Usully, the tems d e i tems of, hece the geel equtio fo the iomil seies epsio is: ( ) ( )...( ) f ( ) f f... f...!! The epsio is oly vlid fo the ge of vlues such tht f() < - < f() <. A simil epsio is ville i MF. Useful fomuls: whee = - By usig simple sustitutio, these fomuls c e used fo y epsio ivolvig powe of - f o f whee the sustitutio is = f() The geel equtios fo these epsios e thus: f f ( ) f ( ) f ( ) f ( )... f f ( ) f ( ) f ( ) f ( )... These epsios e vlid fo f() < - < f() <. 009 M Teo
3 Summy: Biomil Epsio Specil () Fom the epsio of f f ( ) f ( ) f ( ) f ( )..., studets should oseve tht the sigs i the epsio e ltetig. This is sometimes clled ltetig seies. Fo this pticul epsio, the epessio fo the powe tem is ) f ( ) I geel, fo ltetig seies, studets my simply use fcto (. () o the sig. This is vey useful fo this chpte s well s fo summtio d APGP. ( ) to lte () Most epsios usig (*) e i scedig powes of, which lso implies is smll. HOWEVER, thee e questios tht equie epsio i descedig powes of o questio simply sttes tht is lge. Fo emple, The epsio of.. cosideed smll. is i scedig powes of d hee, is If isted, the questios sttes tht is lge OR equies epsio i descedig powes of, studets e equied to pefom the followig steps:.. fctoise efoe epdig... whee the seies is ow i descedig powes of. Now, the epsio is oly vlid if i. e. o The logic fo fctoizig is tht if is lge, must the e smll! ) Applictio of Biomil Seies As studets my hve ledy foud out, iomil seies is ifiite seies. Usully questios equie studets to epd up to mimum of tems (o util the tem). Also, sice c e sustituted with y umeicl vlue, iomil seies epsio c e used s ppoimtio to ceti vlues. 009 M Teo
4 Summy: Biomil Epsio Sometimes the vlue to e used fo the sustitutio is povided i the questio. I the evet tht studets must detemie the ppopite sustitutio, you must choose vlue tht is withi the vlid ge of vlues. Studets c use clculto to check if you swes mke sese. Othe Osevtios: Ptil Fctios (PF) Vey ofte, studets e sked to fid the iomil seies epsio of PF. Fidig the iomil seies is usully ot polem fo studets. Studets will howeve ecoute hece o othewise polems tht sk fo the coefficiet of the tem. Tke fo emple, Fo the epsio of detemie its PF., efoe pplyig iomil seies epsio, studets should ( )( ) ( )( ) which c e esily woked out usig cove-up ule. The epsio is thus: ( ) ( )( ) ( ) Hee, we c pply the useful fomuls i Pge to oti the tems of ( ) () () () ()... () () () ()... d ( ) Hece, ( )( ) ( ) ( ) ( ) ( )... ( ) ( ) ( ) ( ).. --(^) 6 if oly st fou tems e cosideed. 009 M Teo
5 Summy: Biomil Epsio To detemie the coefficiet of, studets must modify step (^) i ode to solve: () () () () () () () ()... ( ) ()... () (-) is used to ltete the sig Fom the modified step ( the tem is icluded fo oth epsios, if you did ot see it), the coefficiet of must the e ( ) () () o simply ( ) () y pplyig idices lw. It is thus dvisle to use the oigil epsio to detemie coefficiet of, uless the fil swe povides cle ptte. Fo this pticul emple, the swe of 6 does ot povide ptte to deduce the coefficiet. Specil Sice the esult ivolves two iomil seies epsio, the ge of vlues of must e stisfied y oth, which c oly e possile y the itesectios of the two ges. Fo, the epsio is vlid fo < Fo, the epsio is vlid fo - < Thus the ge of vlid vlues fo is the smlle set ( )( ) Studets MUST pefom this step oce thee e t lest two iomil seies epsios. 009 M Teo
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