.8,.9: Taylor ad Maclauri Series.8. Although we were able to fid power series represetatios for a limited group of fuctios i the previous sectio, it is ot immediately obvious whether ay give fuctio has a power series represetatio. Which fuctios have a power series represetatio, ad how ca we fid such a represetatio? Suppose that f is ay fuctio that ca be represeted by a power series. 4 f = c + c( a) + c( a) + c( a) + c4( a) + a < R First otice that if = a, all terms after the first term are ad we have f a = c Differetiatig f, we have f = c 4 + c a + c a + 4c4 a + 5c5 a + a < R Agai, lettig = a, all terms after the first term are ad we have f a = c Differetiatig f agai, we have f = c 4 + c a + 4 c4 a + 4 5 c5 a + 5 6 c6 a + a < R Agai, lettig = a, all terms after the first term are ad we have f a = c Differetiatig f oce more, we have f = ( ) c 4 + 4 c4 a + 45 c5 a + 456 c6 a + 567 c7 a + a < R Agai, lettig = a, all terms after the first term are ad we have f a = c =! c So, here is the patter: substitutig = a i the th derivative ( ) =! f ( ) to obtai f a c Solvig this equatio for the coefficiet we have ( f ) ( a) c =! This formula is valid eve for = because! = ad if we adopt the otatio f = f.
Theorem.8. If f has a power series represetatio (epasio) at a, that is, if, a < R = = ( ) f c a the its coefficiets are determied by the formula f a c =! Puttig these coefficiet values back ito the series we see that if f has a power series epasio at a, the it must have the form = ( ) ( a) f f = a! f a f a f a = f ( a) + a + a + a +!!! The series i this theorem is called the Taylor series of the fuctio f at a (or about a, or cetered at a). Whe a =, the series becomes ( f ) f f f f = = f + + + + =!!!! which is called the Maclauri series of the fuctio f. Eample : Fid the Maclauri series of the fuctio f = e ad its radius of covergece.
WARNING!! We must be careful at this poit, however. Accordig to the theorem earlier, if epasio at, the it must be e =. However, we have ot demostrated that =! have a power series represetatio..8. e has a power series e really does With what coditios is a fuctio equal to the sum of its Taylor series? As with ay coverget series, this meas that f ( ) is the limit of the sequece of partial sums. For a Taylor series, these partial sums are ( i f ) ( a) i T = ( a ) i= i! ( f ( a) f ( a) ) f ( a) = f ( a) + ( a ) + ( a ) + + ( a )!!! T is called the th-order (or th degree) Taylor polyomial of f at a. For the previous eample f = e, we have the followig Taylor polyomials at (also called Maclauri polyomials) for =,, : = + T = + + T! So, f ( ) is the sum of its Taylor series if f T Taylor s Theorem: = lim. T = + + +!! If Theorem f has derivatives of all orders i a ope iterval I cotaiig a, the for each positive iteger ad for each i I, the If f = T + R, where T is the th-degree Taylor polyomial of f at a ad f ( a) f ( a) f ( a) f ( a) f = f ( a) + ( a ) + ( lim ar = ) + ( a ) + + ( a ) + R,!!!! for a < R, the f is equal to the sum of its Taylor series o the iterval a < R. ( + ) f () c + where R = ( a) for some c betwee a ad. ( + )! R is called the remaider of the Taylor series ad R f T, the T f ad the Taylor series coverges to f ( ). To show that R =. If R as lim = for a specific fuctio f, we ofte use the followig.
.8.4 Remaider Estimatio Theorem (Taylor s Iequality): If ( + ) () f t M for all t betwee ad a, iclusive, the the remaider R series satisfies the iequality M R a! ( + ) +. of the Taylor Eample : Fid the Maclauri series for f = cos. Eample : Fid the Maclauri series for the fuctio f = si. Istead of computig derivatives (which would be messy), take the Maclauri series for si ad multiply by.
.8.5 f = Eample 4: Fid the Taylor polyomials of orders,,,, ad the Taylor series cos cetered at π. (Put aother way: geerated by cos f = at π ) Eample 5: Fid the Taylor polyomials of orders,,,, ad the Taylor series f = + 5+ cetered at f at.) 4. (Put aother way: geerated by
Importat Maclauri Series.8.6 e = = + + + + (,) = = = = + + + +!!!! (, ) + 5 7 si = = + + (, ) = ( + )!! 5! 7! 4 6 cos = = + + (, ) = ( )!! 4! 6! = = + + + 5 7 ta = + 5 7 [,] d. Eample 6: Evaluate si ( ) si Eample 7: Evaluate lim with a series.