Roerto s Notes o Ifiite Series Chpter : Covergece tests Sectio 4 The limit compriso test Wht you eed to kow lredy: Bsics of series d direct compriso test. Wht you c ler here: Aother compriso test tht does ot require provig iequlity with required directio. Both sttemets of the direct compriso test re oe-wy oly, tht is, we c use them oly whe we c prove iequlity i certi, fvourle directio. As oted efore, this is ot lwys esy or eve possile. But there is esier wy. Strtegy for the limit compriso test Assume tht d re two sequeces tht re evetully positive. I tht cse If is coverget d lim is coverget. L, the If is diverget d lim L 0, the is diverget. The proof of oth sttemets is somewht techicl i its detils, so I will omit it here. However, its sic ide is to show tht, if the required limit is fiite, the the terms of the two series hve the sme reltive size d hece the series hve the sme ehviour. Notice tht oth sttemets use the sme limit d differ oly i the oe cse they exclude: For covergece the limit cot e ifiite. For divergece the limit cot e 0. Otherwise, if 0 L, the oth series coverge, or they oth diverge. The ig questio relted to the limit compriso test is, give series, how do we determie other series to which we c compre it? It turs out tht we c use rough versio of the lw of the jugle. Ifiite Series Chpter : Covergece tests Sectio 4: The limit compriso test Pge
Strtegy for choosig echmrk series I order to pply either the direct or limit comprisos test to series whose sequece is evetully positive, compre it to the series otied y elimitig from the formul of the give series ll the terms of lower power d ll irrelevt coefficiets. Time for exmples. Exmple: By elimitig ll the terms of lower power, s well s the coefficiets, which do ot ffect the covergece, we try to compre this series to the series: This is p-series with p d hece coverget. Sice we kow the properties of this series, we use the limit compriso test d oti: / lim lim lim Sice the limit is fiite, the give series coverges. / Exmple: We c try the limit compriso test y comprig the give series to the oe otied y elimitig terms of lower power d the coefficiet, mely: Sice the ltter is p-series with compute the required limit: p, it will diverge. We ow lim lim lim lim Sice this limit is fiite, the two series hve the sme ehviour d therefore our origil series is lso diverget. Exmple: t This time we hve o low powers, ut we rememer tht the lier pproximtio of y t x er the origi is y x, so we c try to simply drop the tget d compre to the hrmoic series, which we kow to e diverget. The limit of the test provides: Therefore our series is lso diverget. t si lim lim cos Ifiite Series Chpter : Covergece tests Sectio 4: The limit compriso test Pge
Summry The limit compriso test llows us to compre give series to other, kow oe without requirig to check iequlity d through method tht works pretty much (ut ot etirely) oth wys. Commo errors to void Shrpe your skills i selectig suitle series to which to compre the give oe, or you my ot get stisfctory coclusio. Lerig questios for Sectio S -4 Review questios:. Descrie how the limit compriso test for series works.. Descrie the commo wy to use the limit compriso test icorrectly.. Expli the dvtge of the limit compriso test over the direct compriso test. Memory questios:. If d re two positive series, d lim sy tht the two series oth coverge or oth diverge? c whe c we. If d out the covergece of the two series? re two positive series, d lim 0 wht c we sy. If d re two positive series, d lim sy out the covergece of the two series? wht c we Ifiite Series Chpter : Covergece tests Sectio 4: The limit compriso test Pge
Computtio questios: I questios -, use the limit compriso test to ssess the covergece or divergece of the give series. The see if other test c lso e used d, if so, check tht the coclusio of the other test is the sme.. 4.. 6 4 6. 7 9. /4 0. 4.. 4. ( ) 7. 8. 4 6.. Theory questios:. Whe is it dvisle to use compriso test?. Why is the limit compriso test preferle to the direct oe i geerl?. I order to use compriso test, do we eed the series to cosist of positive terms, or of evetully positive terms? 4. Wht ssumptio is ehid oth compriso tests?. For wht limitig vlues is it possile tht the limit compriso test is icoclusive? 6. If you wted to pply the limit compriso test to the series 6 6, to which series would you compre it? Ifiite Series Chpter : Covergece tests Sectio 4: The limit compriso test Pge 4
Proof questios:. Show tht if lim, the the series diverges. Wht questios do you hve for your istructor? Ifiite Series Chpter : Covergece tests Sectio 4: The limit compriso test Pge
Ifiite Series Chpter : Covergece tests Sectio 4: The limit compriso test Pge 6