Topic 9 - Taylor ad MacLauri Series A. Taylors Theorem. The use o power series is very commo i uctioal aalysis i act may useul ad commoly used uctios ca be writte as a power series ad this remarkable result has give rise to may importat results i calculus. Taylors Theorem is states that i () is a well behaved uctio the it ca be represeted by the ollowig power series. () = () = c ( a) where c = () (a) c + c( a) + c2( a) 2 + c c2( a) +.. This power series will coverge or all values o with a < R, or some value R. The Taylors power series o a uctio is usually writte i the orm below ad is called the Taylor series o the uctio at a.! () ( ) ( a)! () = (a) + (a) ( a) + (a) ( a) 2 + + (a) ( a) + + v (a) ( a) 4 + Deiitio: The MacLauri s Series is the special case o Taylors Series with a =. () ()! ( ) '' () (4) '() () 2 () () 4 = ()... This power series will coverge or all values o with < R, or some value R. MacLauris series is a commoly used versio or the power series o a uctio as it teds to geerate simpler polyomials. Whe we state that a uctio is well behaved we mea that it is cotiuous ad all its derivatives eist at least at oe poit = a or a Taylors series ad at = or a MacLauris series to eist. Here is a list o the most commo MacLauri series.. 2. e!. si (2 )! 4. cos 2 (2)! 5. ta 2 (2 ) = + + 2 + + 4 + 5 + = + + 2 + + 4 + 2 = + 5 5! 7 7! + = 2 + 4 6 6! + 8 8! + = + 5 5 7 7 + Page
B Fidig MacLauris ad Taylors series. Eample : Fid the MacLauris series (MacLauris Epasio) or the uctio () = e. Solutio: We start by idig a suitable umber o derivatives at a = i order to detect a geeral patter. () = e () = e = () = e () = e = () = e () = e = () = e () = e = v () = e v () = e = MacLauris epasio is () = () + () + () 2 + () + v () 4 + () = + + 2 + + 4 + () = + + 2 + + 4 + So the power series () = e ()! ( )!! Page 2
Eample 2: Fid the Taylors series (Taylors Epasio) or the uctio () =si. at a = π We start by idig a suitable umber o derivatives at = i order to detect a patter. () = si (π) = si π = () = cos (π) = cosπ = () = si (π) = siπ = () = cos (π) = cosπ = v () = si v (π) = si π = v () = cos (π) = cosπ = The geeral patter here is very simple whe it is a odd derivative the umber is or ad whe it is a eve derivative it is zero. So we eed a geeral rule that oly geerates values or the odd derivatives ad igores the eve derivatives. Odd umbers ca be geerated by usig 2 so the odd (2 ) th 2 - derivative will have coeiciet ( ) So (2 ) (π) = ( ) 2 The Taylors epasio or () = si about = π is () = () = () = (π) + (π) ( π) ( π) + (π) + + ( π) ( ) (2 )! 2 ( π) 2 + (π) + ( π)5 5! + ( π) + v (π) ( π) 4 + Page
Eample : Fid the Taylors series (Taylors Epasio) or the uctio () =. at a = We start by idig a suitable umber o derivatives at = i order to detect a geeral patter. () = 2 = () = () 2 = () = 2 2 () = 2 () 2 = 2 () =. 5 2 2 2 () = 2. 2 () 5 2 =. 2 2 () = 2. 2. 5 2 7 2 () = 2. 2. 5 2 () 7 2 =..5 2 v () = 9.. 5. 7 2 2 2 2 2 v () = 2. 2. 5 2. 7 2 () 9 2 =..5.7 2 4 The geeral patter here is quite comple so the best strategy is to look at idividual patters withi the result. The irst patter to otice is that the sequece o values alterates rom positive to egative to positive etc this is geerated by usig ( ) or ( ) + i this particular case we have that ( ) works. The et patter is that the deomiator o the derivatives has powers o 2, so that the th derivative uses.. The last patter is the umerators which cosist o the product o odd umbers, so or 2 eample the 4 th derivative uses the product o the irst 4 odd umbers..5.7 so the th derivative will use the product o the irst odd umbers =..5.7.9.(2 ) What we do ow is to try ad get a geeral ormula or this product...5.7.9.(2 ) =.2..4..2 =.2..4..2 2.4.6.8. 2 2 (.2..4.5..) = (2)! 2! Puttig this all together we get () () = ( ) (2)! 2 2! epasio about =. () = () + () ( ) + () = ( ) (2)! 2 2! ( a) 2 + () so we have the ollowig Taylors ( ) + v () ( ) 4 + () =. ( ) + ( 2 2 2.() 2 2 () )2 +..5 ( 2 () ) +..5.7 ( 2 4 () )4 + () = ( ) () ( )! (2)! (!.2! ) 2 (2)! ( ) 2 2 2 (!) Page 4
Eample 4: Fid the MacLauris series (MacLauris Epasio) or the uctio () = e. Solutio: We use a shortcut here sice we kow that e! we ca id e = =!! 4 5 6 7...! 5 6 7 4 =... 2 6 24 Eample 5: Fid the MacLauris series (MacLauris Epasio) or the uctio () = si Solutio: We use a shortcut here sice we kow that si (2 )! 2 we ca id si (2 )! 2 (2 )! 2 = 2 + 4 5! 6 7! + Practice :. Fid the MacLauri power series or the ollowig uctios. (a) () = si (b) () = l( + ) (c) () = cos 4 2. Fid the Taylors series at or () = si at a = π 2. Fid the Taylors series at or () = l at a = e 4. Fid MacLauri series or the ollowig uctios usig a shortcut method. (a) () = 4 e (b) si( 2 ) (c) si(4) Page 5
C. Fidig the Radius o Covergece o a MacLauri or Taylors Series. To id the radius o covergece o a MacLauris or Taylors series we will use the ratio test as i previous eamples. Eample : What is the radius o covergece or the MacLauris series () = si( 2 ). Solutio: First we eed to id the MacLauris epasio or (). We ca use the shortcut method as we already kow that y = si (2 )! 2 so the MacLauris epasio or si 2 ( (2 )! (2 )! Net we use the ratio test. lim a + a = lim Sice the lim a + a coversio is R = 42 2 ) 2 ( ) + 4+6 (2+)! ( ) 4+2 (2+)! ( )+4+6 = lim (2+)! ( ) 4 (2+2)(2+) = lim = (2+)! ( ) 4+2 = < the series will coverge or all values o ad its radius o Page 6
Deiitio: The th - degree Taylor Polyomial o at a is a polyomial o degree deied as the irst (+) terms o its Taylors Series (icludig those terms which are zero) T() = k ( ) k ( a) k! ' T() = ( a) '' ( a) 2 ( a) ()... ( a)! ( ) Deiitio: The remaider series R() is deied as the power series ormed by usig the terms that are let rom the Taylors series so it will start at the (+) th power o ad is deied as. R () = k ( ) k ( a) k! ( 2) ( ) 2 = ( a) ( a)... ( a)... ( )! ( 2)! ( )! Aother way to look at this is that R()= () T(). These deiitios lead us to the result that () = T() + R() or all values o with a < R, or some value R provided that lim R () = or ay give value o iside the radius o covergece. It ca also be show that I (+) () M or all values o with a d the R () M( a)+ or all values o with a d (+)! This essetially meas that the remaider i a Tailors series is bouded by some multiple o the (+) th term. Eample 2. Use the MacLauris series or si usig T5() to estimate si(.2) ad give the error boud o this. Solutio: The Taylors series or th degree is T() = ' ( a) '' ( a) 2 ( a) ()... ( a)! ( ) The MacLauris series or () = si or th degree uses a = ad is deied as T5() = T5() = '() 5 5! () () () 5! '' () (4) (4) 2 4 5 ( ) Page 7
We ow eed to id a umber M that has the property (+) () M (6) () M cos M cos or all values o with a d or all values o with.2 d or all values o with.2 d or all values o with.2 d We ow use the remaider theorem which states that we use d =.2 as a choice or d. Sice we oly wish to id oe speciic value o the uctio we could make d very small (almost zero). R () M+ or all values o with a d (+)! R 5 () 6 or all values o with.2.2 6! R 5 () (.2)6 or all values o with.2.2 6! R 5 (). 5 or all values o with.2.2 Net we calculate T5(.2) = 5 (.2) (.2).2 =.972274 5! So we ca coclude that si(.2) =.9 72 2 ±. 5 Usig a calculator we i act get si(.2) =.9 722 7 ow thats accurate!!!!! Note: We used =.2 as that is the largest value o i the iterval.2.2 ad so is the largest possible error. Page 8