Math Fall 7 Lab Represetig Fuctios as Power Series I. Itrouctio I sectio.8 we leare the series c c c c c... () is calle a power series. It is a uctio o whose omai is the set o all or which it coverges. I sectio. we leare the series ar a ar ar ar... ar... a, () is a geometric series that coverges to the sum, a s, i r. r I we let c i (), the power series becomes the geometric series where a = a r. It ollows, i, the power series coverges to the sum, be represete as a power series: s. Thus, we see a uctio that ca...... whose omai is () II. Creatig Power Series Represetatios or Other Fuctios By maipulatig the epressios i equatio (), we ca represet other uctios as powers series. This maipulatio iclues algebra, substitutio, ieretiatio a itegratio. The ollowig three eamples emostrate how this works. Simpler eamples are i your tetbook o pages 77-75.
Eample : Epress as a power series. At irst glace, this shows o resemblace to the uctio i () but we use algebra to maipulate a create a substitutio or i the geometric series. Geerally, we wat the orm: ) ( h g g h. For the uctio above we actor out rom the umerator a rom the eomiator. The, replace i equatio () with. So, The power series represetatio or is. Rules or ieretiatio a itegratio o uctios ca be applie i power series problems. A. Usig the quotiet rule or ieretiatio we see ) ( ) ( () () We ca also ieretiate a power series term-by-term to obtai:...... B. Usig the ieiite itegral C l alog with substitutio we see C l We ca also itegrate a power series term-by-term to obtai:...... l C
Eample : Fi the power series represetatio or. Notice the ierece betwee this uctio a the oe i Eample. The etire eomiator is square here which provies a clue as to what type o maipulatio is eee. The quotiet rule or ieretiatio results i squarig the eomiator so we kow the uctio we must ieretiate has (+) i the eomiator. The i the umerator ca be actore out a igore temporarily. We see maipulatig () requires a little ivestigatig but a goo startig poit is to ieretiate /(+): () () Now that we kow what the erivative looks like, we ca write () i terms o this erivative. Why o we go through all these maipulatios? We ca ow create a power series or /(+) a ieretiate it to create aother power series. We the iclue the actor i the power series. The power series represetatio or is. There are a ew importat poits worth otig.. The erivative o a power series IS aother power series! The same is true or the itegral o a power series.. The raius o covergece remais the same whe a power series is ieretiate or itegrate but the iterval o covergece might chage.
Eample : Fi the power series represetatio or l. Oce agai, there is a coectio betwee /(+) a the give uctio. Itegratio provies this coectio so a goo startig poit i creatig the power series or () is to itegrate /(+). C l Now that we see the itegral oly iers rom () by a actor o /, we ca write () i terms o this itegral. l l We ca ow create a power series or /(+) a itegrate it to create the power series or (). We the iclue the actor i the power series. The power series represetatio or l is. III. Approimatig a Fuctio usig a Power Series We leare i. that ay partial sum s ca approimate the sum o a coverget series. How oes this relate to our power series represetatios o a uctio? Suppose we wat to approimate the sum o the coverget geometric power series... at (remember, this series oly coverges or ). Usig the partial sum, we get 5 s s. We kow the eact sum is s sice... Calculatig s amouts to summig the irst two terms i the series. So, i we o ot i the value o, the approimatio o the sum is a liear equatio i. That is or.
5 Below are the graphs o a the liear approimatio ( ). Notice, the lie provies a goo approimatio o the uctio or values o that are very close to =. I Math, we leare a ormula or the liear approimatio o a uctio ear a poit = a. Give a uctio, i a the ( ) ( a) ( a)( a) () Usig our uctio a =, we use this ormula a get the same liear approimatio we obtaie rom the power series. This leas to a ew questios. Questio. Sice s provies a better approimatio tha s or the sum o a coverget series, oes aig aother term i the power series also show a better approimatio (graphically)? Aig the et term o the power series or a approimatio yiels this quaratic approimatio is better.. We ca see
Questio. Is there a coectio to the liear approimatio ormula i the bo above a the power series represetatio o a uctio? I so, oes this coectio ete to the quaratic approimatio as well? The aswer is yes a we will ivestigate. The liear approimatio ormula, satisies the ollowig coitios: Coitio : At = a =, a the taget lie meet. Coitio : At = a =, a the taget lie have the same slope (i.e. their erivatives are equal at =.) The geeral ormula that satisies these coitios (or a = ) is ) ( (5) The quaratic approimatio ormula, satisies the ollowig coitios: Coitio : At = a =, a the taget parabola meet. Coitio : At = a =, a the taget parabola have the same slope (i.e. their erivatives are equal at =.) Coitio : At = a =, a the taget parabola have the same cocavity (i.e. their seco erivatives are equal at =.) The geeral ormula that satisies these coitios (or a = ) is ) ( () Questio. Aig more terms to s yiels a better approimatio o the sum; oes aig more terms i the power series yiel a better approimatio to the uctio? Does the geeral ormula or the approimatig polyomial cotiue to have a patter? The aswers are yes a yes. Each higher egree polyomial that approimates to the taget lie a the taget parabola as well as: (COMPLETE THE EXERCISE PROBLEMS) will satisy the previous coitios aalogous The th erivatives o a the taget th egree polyomial are equal at =. The geeral ormula that satisies these coitios (or a = ) is ()... (7)!!! This series is calle the Maclauri Series. Notice the ormula allows us to i power series represetatios or other uctios.
7 Math Fall 7 Eercises Lab Name: Sectio: Score:. (a) I Eample why oes the series begiig at = chage to =, as show? = (b) I Eample why oes the series begiig at = chage to =, as show? = (c) It is sai (i class, i the tetbook a i this lab) that... but there seems to be a problem. For eample,... 8. Eplai what your istructor, the tetbook a this lab mea whe they say..... Fi the power series represetatio or by maipulatig equatio ().. Fi the power series represetatio or by maipulatig equatio ().
8 by maipulatig equatio (). Hit: usig Laws o Logs the create two power series to combie. Your ial. Fi the power series represetatio or l begi by simpliyig aswer shoul be a sigle power series. 5. (a) Complete the table the i the Maclauri Series or usig equatio (7).
9 () (b) Why is this power series iite istea o iiite? (c) O the cooriate plae below, sketch the graphs o as well as the liear a quaratic approimatios o at =. (Use your graphig calculator.)