Maclaurin and Taylor series

Transcription

1 At the ed o the previous chapter we looed at power series ad oted that these were dieret rom other iiite series as they were actually uctios o a variable R: a a + + a + a a Maclauri ad Taylor series + I a power series is coverget, we ca iput a value o withi its iterval o covergece ad get a iite value or the sum; that is a value or the uctio at that poit. But what uctio would we be evaluatig? Could it be a uctio with which we are already amiliar, such as e, l or si? Ca we relate power series to uctios such as these? We ca already mae a start at aswerig this; we ow rom chapter 7 o the courseboo that, or eample, the uctio ) ( + ) has a power series epasio sice it is the sum to iiity o a geometric series o irst term ad commo ratio ; that is: 5 ( ) < I it is possible to epress the uctio ) ( + ) as a power series, it should be possible to epress other uctios as power series. I this chapter we loo at methods or idig power series represetatios or uctios ad see how we ca use them to mae approimatios or these uctios ad to evaluate limits. I this chapter you will lear: how to id power series i or certai uctios how to id the error i taig oly the irst terms o a power series as a approimatio to orm power series o composite uctios usig stadard results to id power series i a, (where a R is a costat) or certai uctios to use power series to id limits o uctios ( ) o the orm lim a g. See Sectio 7G o the courseboo or sum to iiity o a geometric series. A Maclauri series As we are able to maipulate, dieretiate ad itegrate power series (withi their radius o covergece), it seems a good idea to epress other types o uctio as series so that we ca aalyse them i the same way. Cambridge Mathematics or the IB Diploma Higher Level Cambridge Uiversity Press, Optio 9: Maclauri ad Taylor series

2 To this ed, cosider a power series with radius o covergece R > ad suppose we deie a uctio: ) a + a + a + a + < R a Lettig we immediately have: a Sice we ca dieretiate the power series or R, we have: The otatio or the irst th derivative o a uctio, ( ) ) was itroduced i Sectio 6D o the courseboo. Dieretiatig agai: Ad agai: a + a + a + a + a ) a + ( )a a + ) ( a + a ( ()! ) ( ) a + ( )aa + a ()! Cotiuig i this way we will get: () ) + + () () + +!!!! KEY POINT. ()! The power series ) ) + ( ( ( ) + +!! is ow as the Maclauri series o the uctio ). It eists provided that () eists or all N. I Sectio D we will see that it is useul to be able to id series epasios about poits other tha. As we are evaluatig all o the derivatives at, we say that the Maclauri series o a uctio is cetred at, or tal o the series epasio aroud. We caot id a Maclauri series or every uctio we have met so ar (or eample ) l does ot satisy the above coditio o () eistig or all N; ideed it does ot eist or ay N) but we ca id Maclauri series or may amiliar uctios. Cambridge Mathematics or the IB Diploma Higher Level Cambridge Uiversity Press, Optio 9: Maclauri ad Taylor series

3 Wored eample. Fid the Maclauri series or: (a) ) e (b) g si Give your aswers i the orm a We eed to establish the irst ew derivatives o () ad evaluate them at. We see that clearly ( ) or all { } Use the Maclauri series ormula We eed to establish the irst ew derivatives o () ad evaluate them at. We see that as ( ) there is a cycle o values that repeats cotiually such that ) (a) ( ( ( ) ( ) ( ( ) ( ( ( ) ( So, +!!!! !!!!! !!!! (b) ( ) ( ( ) cos ( ) s ( ) ( ) cos ( 5) ( ) + Use the Maclauri series ormula To orm the geeral term, we ote that this is a alteratig series ad so eed ( ), ad as we oly have odd powers ad actorials, + will geerate them So, +!!!! !!!!! 5! ! 5! 7! + ( ) ( + )! We oud i Eercise 9C, questios (b) (i) ad (b) i this optio that both these power series coverge or all R (although we did ot relate the series to e ad si ). Cambridge Mathematics or the IB Diploma Higher Level Cambridge Uiversity Press, Optio 9: Maclauri ad Taylor series

4 As metioed above, we caot id the Maclauri series or y l. However, we ca or y l + ) as this avoids the problem o () (ad all the derivatives at ) ot eistig. Wored eample. Fid the Maclauri series or ) l ( + ), givig your aswer i the orm a. We eed to establish the irst ew derivatives o () ad evaluate them at. We ca see a patter o alteratig sigs o actorials Use the Maclauri series ormula Agai this is a alteratig series but with the sum startig at, but we eed ( ) + to mae the irst term positive ( ) ( + ) ( ) () ( ( + ) (! ( + ) ( )! ( ) ( )! ( )! 5 ( 5) ( )! So, +!!!! !! 5 +!!!!!! 5! ( ) + eam hit Loo or patters i the irst ew derivatives at to be coidet o how the series will behave. We oud i Eercise 9C, questio (a) that this power series coverges or <. Clearly, to have ay hope o the series covergig to the value o the uctio at, we ca oly tae a value o iside the iterval o covergece o the Maclauri series o a uctio. However, i we ow that a Maclauri series coverges or particular values o, this does ot mea that it coverges to the uctio it was derived rom! We will see i the et sectio how we ca determie whe the Maclauri series o a uctio does ideed coverge to that uctio. Eercise A. Fid the irst our o-zero terms o the Maclauri series or the ollowig uctios. (a) (i) cos (ii) si (b) (i) e (ii) e (c) (i) + (ii) ( ) (d) (i) ta (ii) sec Cambridge Mathematics or the IB Diploma Higher Level Cambridge Uiversity Press, Optio 9: Maclauri ad Taylor series

5 . Fid the irst three o-zero terms o the Maclauri series or ) si. [5 mars]. Show that the Maclauri Series up to the term i or l(+si) is + 6 [6 mars]. (a) Fid the term up to i the Maclauri epasio o ) l( cos ). (b) Use this series to id a approimatio i terms o π or l. [7 mars] 5. (a) Show that the Maclauri Series or ) ) ! 8! (b) Fid the radius o covergece o this series. is 5... ( ) +...! 8 [8 mars] 6. (a) Fid the Maclauri epasio o ) l ( + ), givig your aswer i the orm a. (b) Fid the iterval o covergece o this power series. [9 mars] 7. (a) Fid the Maclauri epasio o + up to the term i. (b) Prove that this power series coverges or <. [9 mars] 8. y Eplai why either o the ollowig ca be Maclauri series o the uctio (): (a) (b) + Cambridge Mathematics or the IB Diploma Higher Level Cambridge Uiversity Press, Optio 9: Maclauri ad Taylor series 5

6 Loo bac at the epressio or the trucatio error o a alteratig series i Sectio 9B (Key poit 9.8) o this optio. B Approimatios to the Maclauri series Have you ever wodered how a calculator or computer ids a value or, say, si.5 or e? Oe possible way is to programme the calculator with the Maclauri series so that whe values o are iputted, it ca simply evaluate this power series. It caot o course use iiitely may terms o such series, so how may terms is eough to give a suicietly accurate aswer? This is similar to the issue or alteratig series i chapter 9 o this optio, where we oud a epressio or the trucatio error at terms o the series; but here, by trucatig the Maclauri series, we are actually determiig a th degree polyomial, which approimates the uctio. KEY POINT. The trucated Maclauri series: () () ( ) !!!! () is reerred to as the th degree Maclauri polyomial, p () o the uctio (). ()! Usig the Maclauri series or si (Wored eample.(b)) to approimate the uctio ear to (here we use.5), we ca see how accurate the irst ew Maclauri polyomials or si are. So, or the irst degree Maclauri polyomial we have si : y y y si si This would give us si.5.5, which is a error o just over %. For the third degree Maclauri polyomial we have si!. 6 Cambridge Mathematics or the IB Diploma Higher Level Cambridge Uiversity Press, Optio 9: Maclauri ad Taylor series

7 y y si y! si! This would give us si , which is oly a.5% error. For the ith degree Maclauri polyomial we 5 have si + :! 5! y y! + 5 5! y si si! + 5 5! This would give us si , which is ow accurate to 5DP. Clearly, our approimatio becomes more ad more accurate the higher the degree o the polyomial; with the ollowig result, we ca quatiy the trucatio error or ay Maclauri series. KEY POINT. For a uctio () or which all derivatives evaluated at eist: ) ()! where the error term R is give by: R ) R c) c + or some c ( + )! ] [ This is sometimes reerred to as the Lagrage orm o the error term. eam hit Although the epressio or the Lagrage error term or Maclauri series does ot appear i the Formula boolet, the more geeral case or Taylor series, eamied i Sectio D, is give i the Formula boolet. Cambridge Mathematics or the IB Diploma Higher Level Cambridge Uiversity Press, Optio 9: Maclauri ad Taylor series 7

8 The proo o this result is beyod the scope o the syllabus but it is possible to uderstad the Lagrage error term by thiig about the particular case o a th order (i.e. costat) Maclauri polyomial where the error term is give by the irst derivative. The Mea Value Theorem states that there must eist some c ] [ such that, ) () c) + () which is eactly the result i ey poit., with. This idea the geeralises to higher order derivatives. I estimatig the error i taig a th degree polyomial to approimate a uctio, we will ot ow the particular value o c, so istead we will id the largest possible error (a upper boud or the error) by taig c to be the worst case possible. Wored eample. (a) Fid a epressio or the error term i approimatig e by its d degree Maclauri polyomial. (b) Give a upper boud to DP o the error whe usig this approimatio to id e.75. First state the d degree Maclauri polyomial usig the series oud above Fid the error, otig that all derivatives o e are e Evaluate at.75 As e c is a icreasig uctio, the largest possible value o c will give the largest possible value (the upper boud) o the error (a) The d degree Maclauri polyomial gives the approimatio e + + with error term R ( ) ec! ( c)! c ] [ (b) Taig.75 we have c e 75 R c 75,.! 7. 5 e 7. 5 R <! c ] [. 89 eam hit For alteratig series (that satisy the appropriate coditios), it is easier to boud the error at a particular value o usig the result or the trucatio error i Key poit 9.8 rather tha the Lagrage error term. 8 Cambridge Mathematics or the IB Diploma Higher Level Cambridge Uiversity Press, Optio 9: Maclauri ad Taylor series

9 We ca use Key poit. to boud errors resultig rom the trucatio o Maclauri series, but we ca also determie or which values o a Maclauri series actually coverges to the uctio rom which it was derived. (Remember, we said at the ed o the last sectio that a Maclauri series does ot ecessarily coverge to the uctio rom which it was derived or all i the iterval o covergece.) Sice ) ()! i we let, it maes sese that ) oly i the error term teds to. KEY POINT. R ()! The (iiite) Maclauri series or () is precisely the same as the uctio () or all where lim ( ) Wored eample. Prove that the Maclauri series or e is valid or all R. We eed to show that lim R ( ) From Wored eample. ( ) e ( + ) e or all Z + Thereore, + ( c ) R ( ) c ( + )! ] [ Simpliy the modulus sig, otig that e c >, + ) > ad + + Notig that e M or some costat M R, we ca apply the algebra o limits + e c ( + )! c + lim R lim e ( + )! + lim e c ( + )! + M lim ( + )! or somem R From chapter 7 Mied eamiatio practice, questio 6 i this Optio, we ow that lim! + Sice lim, ( + )! R lim Cambridge Mathematics or the IB Diploma Higher Level Cambridge Uiversity Press, Optio 9: Maclauri ad Taylor series 9

10 cotiued... eam hit lim is a useul! limit to ow. It ca be quoted i this case but you should also ow how to prove it. Ad sice, clearly, R by the Squeeze Theorem, Hece, e! R lim or all R A similar argumet wors or si or cos. Wored eample.5 Show that the Maclauri series or cos :! ) coverges to cos or all R., or We eed to id t ( t) cos t, so we id the irst ew derivatives ad loo or a patter. There is a cycle o values that repeats cotiually such that t t ) Sice all derivatives will be either ±sit or ±cost, we ow that () t will be oe o these ( t ) cost ( ) sit ( cost ( sit ( t ) cos t ( t ) si t ) ± si t ( + )( ± cost We eed R t Sice ( t) ± si ± cost we ow that ( c) R ( ) c) ( + )!) ( + )! + ( + )! + +! + c + ) + Agai lim! + Sice lim, ( + )! R lim Ad sice, clearly, R by the Squeeze Theorem, Hece, cos ( ) R lim! orall R Cambridge Mathematics or the IB Diploma Higher Level Cambridge Uiversity Press, Optio 9: Maclauri ad Taylor series

11 We see rom Wored eamples. ad.5 that the values o or which the Maclauri series o the uctios e ad cos are equal to the uctios themselves coicide eactly with the itervals o covergece o the respective power series. This will always be the case or the uctios we meet, but we still eed to be able to show this or each particular uctio. A eample o a uctio that is ot equal to its Maclauri series is e Ivestigate the Maclauri series ad graph o this uctio. Eercise B. Fid a upper boud o the error whe usig the Maclauri polyomials o give degree to approimate the ollowig uctios or 5: (a) (i) cos p, (ii) ta p, (b) (i) e, p (ii) e, p (c) (i) +, p ( ) (ii), p ( ). (a) Fid the secod degree Maclauri polyomial or e. (b) For what values o > will this polyomial approimate e to withi.? Give your aswer to DP. [6 mars]. Usig the Lagrage error term, id a upper boud o the error whe usig: 5 (a) + to estimate si or π π! 5! 6, 6 (b) + to estimate ta or π π, [8 mars]. For, show that e <!! 8 [6 mars] 5. Let ) e + e (a) Fid the ourth degree Maclauri polyomial or (). (b) Hece id a approimatio to, givig your aswer i the orm a b, where a, b Z.. (c) Usig the Lagrage error term, place a upper boud o the error i this approimatio. [ mars] 6. (a) Fid the irst three o-zero terms o the Maclauri series or arcta. (b) Show that the series ( + ) + coverges or < ad that it coverges to arcta. [ mars] Cambridge Mathematics or the IB Diploma Higher Level Cambridge Uiversity Press, Optio 9: Maclauri ad Taylor series

12 7. What degree Maclauri polyomial o e must be tae to guaratee a estimate o e to withi 6? [6 mars] 8. Show that the Maclauri series or si coverges to the uctio or all R. [8 mars] 9. (a) Use the Lagrage orm o the error term to boud the error ivolved i approimatig l ( ) at 5 by (b) By otig that R ( 5) give a improved boud o the error. [9 mars]. (a) Fid the irst three terms o the Maclauri Series or +. (b) How may terms o the epasio o + are eeded to guaratee idig a value o. 8 accurate to withi 6? [ mars]. Show that the Maclauri series or + coverges to the uctio or <. [ mars] C Maclauri series o composite uctios The ollowig stadard Maclauri series (which we have already met), appear i the Formula boolet: KEY POINT.5 e + + +! l( + ) + si 5 +! 5! cos +!! 5 arcta These ca be used to id Maclauri series o more complicated uctios. Sometimes this is straightorward. Wored eample.6 Usig the Maclauri series or cos, id the series epasio o cos( ). We just eed to substitute ito the ow series or cos ( ) + 6 cos( )!! 6! Cambridge Mathematics or the IB Diploma Higher Level Cambridge Uiversity Press, Optio 9: Maclauri ad Taylor series

13 Ote this will ivolve idig two separate Maclauri series ad the combiig them. Wored eample.7 Usig the Maclauri series or si ad e, id the series epasio o e si as ar as the term i. We start by substitutig the series or si, oly goig as ar as the term We ow use the series or e oly goig as ar as ad the epad e + si! e! ee !!! eam hit It is much quicer to orm Maclauri series i this way so where possible i the eam combie ow series. We ca also use results o itervals where the epasio coverges to the uctio (as we ow these will be the itervals o covergece o the uctios we will meet) to id the values or which the epasio o composite uctios are valid. Wored eample.8 Fid the Maclauri series up to the term i or l the epasio is valid. ad state the iterval i which We ow the series epasio or l( ) so rearrage the origial uctio ito separate uctios i this orm Now id the series epasio or each separately l l ( ) l( ( ) l( ) ) l( ( ) l( ) l + l( ) l l + ) ( l ( ) + ( ) ) ( Cambridge Mathematics or the IB Diploma Higher Level Cambridge Uiversity Press, Optio 9: Maclauri ad Taylor series

14 cotiued... Ad ially put everythig together We cosider the iterval o validity separately or each uctio ad the ote that or both to be valid we eed the smaller iterval + + l l l l l Sice l is valid whe <, l is valid whe < i.e. whe < l + is valid whe < i.e. whe < Thereore, l is valid whe < Eercise C I this eercise you ca assume all the stadard Maclauri series results give i the Formula boolet.. Fid the irst our o-zero terms o the Maclauri series or: (a) (i) si( ) (ii) cos( ) (b) (i) l (ii) l (c) (i) e (ii) e. By combiig Maclauri series o dieret uctios id the series epasio as ar as the term i or: (a) (i) lsi (ii) lcos e (b) (i) si (ii) + (c) (i) l( + si ) (ii) l( si ). Fid the Maclauri series as ar as the term i or e si. [ mars]. Show that + +. [5 mars] 6 Cambridge Mathematics or the IB Diploma Higher Level Cambridge Uiversity Press, Optio 9: Maclauri ad Taylor series

15 5. (a) Fid the Maclauri series or l( + ), givig your aswer i the orm a (b) State the iterval o covergece o the power series. [6 mars] 6. (a) Fid the irst two o-zero terms o the Maclauri series or ta. (b) Hece id the Maclauri series o e ta up to ad icludig the term i. [6 mars] 7. (a) By usig the Maclauri series or cos, id the series epasio or cos up to the term i. (b) Hece id the irst two o-zero terms o the epasio o l sec statig where the epasio is valid. (c) Use your result rom (b) to id the irst two o-zero terms o the series or ta. [8 mars] 8. (a) Fid the irst terms o the Maclauri series or ) l[ ] (b) Fid the equatio o the taget to ) at. [ mars] 9. (a) Fid the Maclauri series or l + statig the iterval o covergece o the power series. (b) Use the irst three terms o this series to estimate the value o l, statig the value o used. (c) Provide a upper boud o the error i your approimatio usig the Lagrage error term. (d) Reie the upper boud o the error by cosiderig the error as a geometric series.. Usig the stadard result or e, orm a series or e. Why is this ot a valid epasio? Does e have a Maclauri series? [ mars] D Taylor series We have see that a Maclauri series is valid i a iterval cetred o ad that close to this poit, ote just a ew terms o the series are eeded to give a very good approimatio o the uctio. However, urther away rom, eve withi the iterval where the epasio is valid, you ca eed may more terms o the series to get a reasoable degree o accuracy. For eample, we oud above that the 5th degree Maclauri polyomial or si approimated si.5 correct to 5DP but Cambridge Mathematics or the IB Diploma Higher Level Cambridge Uiversity Press, Optio 9: Maclauri ad Taylor series 5

16 y i we try to use the same polyomial to approimate a value urther away rom, say at, we would get si which is correct to oly DP. The urther away rom we go the worse this gets. y y si.5 y si y + 5! 5! y! + 5 5! y For uctios such as si that have a Maclauri series valid or all R,we ca overcome this diiculty by simply taig higher ad higher order Maclauri polyomials to get the desired degree o accuracy. For other uctios, such as l whose series is oly coverget or <, we caot mae ay reasoable approimatio at all outside this iterval, o matter how may terms o the polyomial we tae. y zoe o covergece y y l( + ) zoe o covergece y l( + ).5 To overcome these problems we will try to covert the Maclauri series epasio to a power series cetred o a geeral poit a rather tha. I this way we will be able to mae a reasoable polyomial approimatio o a uctio ear to ay give poit. So, startig with the Maclauri series: g g g g + () + () g () + +!!! ad lettig g + a so that g( ) ( ) + a ) we have ) 6 Cambridge Mathematics or the IB Diploma Higher Level Cambridge Uiversity Press, Optio 9: Maclauri ad Taylor series

17 a a a) a) a) ) +!!! Ad ially replacig + a with : ) a) + ) a a) a) ( a ) + ( a ) + ( a) +!!! This is ow as the Taylor series. All o the results we have used or Maclauri series geeralise i this way. KEY POINT.6 Taylor s theorem For a uctio () or which all derivatives evaluated at a eist: ) a ) + a )( a ) + + where the Lagrage error term R R a ) ( a ) + R, a )! ( a) is give by,, a ) c)! ( ) ( + ) + a orsome c ] a, [ This is the orm i which this error term is give i the Formula boolet; o course to reduce this to the error term o a Maclauri series we eed oly let a. Wored eample.9 (a) Fid the Taylor series epasio or ) l aroud the poit. (b) Usig the th degree Taylor polyomial as a approimatio or this uctio, id the maimum error or,. Start by idig the irst ew derivatives ad loo or a patter. Here we clearly have ( ) > or Apply the Taylor series ormula with a (a) () ) ( () (! (!! ( ) ()! ( ( )! 5 ( 5) ( )! () () () ()!!!! + ( ) + + ( ) +!!!!!!! + ( ) ( ) + ( ) ( ) ( + ( + + ) ( ) ) ( ) + Cambridge Mathematics or the IB Diploma Higher Level Cambridge Uiversity Press, Optio 9: Maclauri ad Taylor series 7

18 cotiued... Fid the error term or this trucatio ad loo to place a upper boud o this Use the lower boud or c to produce the upper boud o 5 ( ) 5 5c (b) We have: Where, R ( ) ( 5) c )( ) (, ) 5! 5 5 c ( ) 5! ( ) 5 5c 5 < , c, R(, ) Eercise D. Fid the irst three o-zero terms o the Taylor series or each o the ollowig: (a) (i) ta about π (ii) si about π (b) (i) about (ii) e about (c) (i) cosec about π (ii) cot about π 6. Use Taylor s epasio to epress l as a series i, as ar as the term i. [ mars]. Fid the Taylor epasio o e i powers o up to a icludig the term i ( ). [ mars]. (a) Fid a series epasio or si π 6 + up to degree three. (b) Hece estimate the value o si 5 o givig your aswer to sigiicat igures. (c) Provide a upper boud or the error i your estimate. [6 mars] 8 Cambridge Mathematics or the IB Diploma Higher Level Cambridge Uiversity Press, Optio 9: Maclauri ad Taylor series

19 5. Fid a upper boud or the error i usig ( ) ( ) ( ) to estimate or 5,. [5 mars] 6. (a) Fid the secod degree Taylor polyomial at a 8 o ) (b) Show that or 7 9 R (, 8) <. [6 mars] 7. Show that ) has o Maclauri epasio but has a Taylor epasio about ay poit. [5 mars] 8. Let ), R +. (a) Show that l ( ). (b) Fid the Taylor epasio o () about e up to the term i. (c) Fid the maimum value o this polyomial ad coirm that this is the same as the maimum o (). [ mars] 9. Prove that the Taylor epasio o cos about ay value coverges to cos or all values o. [8 mars]. Use Taylor s theorem to show that whe h is small the error i approimatig the derivative o ) at a usig ( + h) ( h) ( ) h is approimately 6 h ( a ). [6 mars] E Applicatios Represetatios o uctios usig Taylor series ca be useul i a umber o cotets. We loo here at two: itegratig a uctio that is otherwise very diicult to itegrate (i ot impossible to itegrate as a stadard uctio) ad idig limits o the orm,, or. For the secod we already have L Hôpital s Rule but sometimes it is easier to do with Taylor series, especially i the irst part o the questio has already ased or a Taylor series! We will loo irst at idig a epressio or the itegral o a uctio that caot be itegrated to give a stadard uctio; the power series method is just about the best way o represetig the itegral i this case. At the ed o the et chapter we will loo at a third applicatio o Taylor Series: idig approimatios to the solutios o dieretial equatios. Cambridge Mathematics or the IB Diploma Higher Level Cambridge Uiversity Press, Optio 9: Maclauri ad Taylor series 9

20 Wored eample. Fid si ) d as a power series, givig your aswer i the orm a Usig the stadard result o the Maclauri series or si we ca id the series epasio or si( ) As this epasio is valid or all R we ca itegrate term by term si si So, + ( ) ( + )! ( ) ( + )! + ( ) ( + )! + ) + d ( + )! + ( ) d ( + )! + ( ) ( + )( + )! orall R orall R Wored eample. I the et Wored eample we loo at a alterative to L Hôpital s Rule or a limit o the orm,. cos + Usig the Maclauri series or cos, evaluate lim. We hope to be able to cacel the i the deomiator beore taig the limit to avoid the situatio So, substitute i the Maclauri series or cos ad simpliy the ractio beore taig the limit 6 cos + +!! 6! 6 cos + + +!! +! ! 8! + 6! 8! Now that the i the deomiator has cacelled, we are ree to let So: cos + lim + lim!!! 6 8! Cambridge Mathematics or the IB Diploma Higher Level Cambridge Uiversity Press, Optio 9: Maclauri ad Taylor series

21 Eercise E. Evaluate usig Maclauri series: (a) (i) lim si (ii) lim cos e e e e (b) (i) lim (ii) lim si cos si si (c) (i) lim (ii) lim cos +. Fid Maclauri series or the ollowig i the orm a : cos (a) d l( + ) (b) d [8 mars]. (a) Fid the irst three o-zero terms o the Taylor series or si( ) about the poit. 9 (b) Usig this series, id lim. [6 mars] si. (a) Fid the Maclauri series or arcta as ar as the term i 7. (b) Hece evaluate lim si arcta. [7 mars] 5. (a) Usig the Maclauri series or l(+) ad cos, show that the Maclauri polyomial o degree or l(cos ) is. (b) Fid the Maclauri polyomial o degree 6 or ta( ). (c) Hece, evaluate lim ta( ) l(cos ). [9 mars] 6. (a) By usig the Maclauri series or l + or otherwise, id a power series epasio o l up to ad icludig the term i. l (b) Hece id lim. [7 mars] 7. (a) By taig the th degree polyomial or e, id a approimatio to: e d (b) Place bouds o the error i this approimatio. [8 mars] si 8. (a) Fid a power series or d i the orm a. (b) How may terms o this series are required to esure a error o less tha 9? [ mars] Cambridge Mathematics or the IB Diploma Higher Level Cambridge Uiversity Press, Optio 9: Maclauri ad Taylor series

22 Summary A Maclauri series is a iiite polyomial which matches all the derivatives o a uctio at zero. The Maclauri Series or ) is give by: ) ) + ( +!! The th degree Maclauri polyomial o the uctio () is give by: ( + ) + + +!!! This is the ull Maclauri series trucated at terms. The error term i usig a th degree Maclauri polyomial is give by: R ) c) + or some c ( + )! ] [ Taylor series geeralise Maclauri series to allow epasio about ay poit, a: a ) ) a ) + a )( a ) + ( a ) a ) + + ( a ) + R (, a)!! where the error term R (, a) is give by R, a) c)! ( ) ( + ) + a) o so e c ] a, [ Taylor series ca be used to provide a represetatio or the itegral o uctios ad to eable limits o the orm,, ad to be oud. Cambridge Mathematics or the IB Diploma Higher Level Cambridge Uiversity Press, Optio 9: Maclauri ad Taylor series

23 Mied eamiatio practice. (a) Fid the irst our derivatives o ) l ( + ). (b) Hece id the irst our o-zero terms o the Maclauri series or (). (c) Usig this epasio, id the eact value o the alteratig harmoic series: + + [7 mars]. Fid the Taylor series epasio o csc i ascedig powers π o up to ad icludig the term i π. (a) Fid the degree 5 Maclauri polyomial o e si. [5 mars] (b) Use the result o part (a) to id e lim si [6 mars]. (a) Fid the Maclauri series up to the term i or ) + (b) Use the Lagrage error term to show that the largest error that could occur whe usig this polyomial to approimate ) or is. [7 mars] 5. (a) Fid the Maclauri series or e, statig the irst our o-zero terms ad the geeral term. t (b) Hece id a Maclauri epasio or t dt. (c) Hece show that + + (!) + 6 (!) +. [ mars] 6. (a) Fid a Maclauri series epasio or e. (b) Hece evaluate e d correct to withi a error o.. [8 mars] 7. Usig Taylor s theorem, show that 8. (a) Show that cos + or all R [6 mars] ( + ) ! Cambridge Mathematics or the IB Diploma Higher Level Cambridge Uiversity Press, Optio 9: Maclauri ad Taylor series

24 (b) Fid the radius o covergece o this power series. (c) Hece id a power series or arcsi statig the values o or which this is coverget. 9. (a) Show that () dt or some uctio. (b) By itegratig a appropriate power series, id the Maclauri series o arcta up to a icludig the term i 7. (c) Hece evaluate lim arcta (d) (i) Use your series (up to the term i 7 ) or arcta to estimate π. (ii) What assumptio have you made? (iii) Fid a upper boud or the error i your estimatio.. The uctio is deied by ) l. (a) Write dow the value o the costat term i the Maclauri series or ). (b) Fid the irst three derivatives o () ad hece show that the Maclauri series or () up to ad icludig the term is: + + (c) Use this series to id a approimate value or l. (d) Use the Lagrage orm o the remaider to id a upper boud or the error i this approimatio. [ mars] [ mars] (e) How good is this upper boud as a estimate or the actual error? [7 mars] ( IB Orgaizatio 8). (a) Usig the series or e, write dow the Maclauri epasio o e i the orm a (b) Show that the Lagrage error term, R () satisies + R ) e + ( + ) i! (c) Hece show that usig 6 terms o the epasio i (a) to approimate e, gives a error o less tha. i the approimatio o / e d (d) How may terms are required to esure the error is less tha 9? [ mars] Cambridge Mathematics or the IB Diploma Higher Level Cambridge Uiversity Press, Optio 9: Maclauri ad Taylor series