Calculus 2 TAYLOR SERIES CONVERGENCE AND TAYLOR REMAINDER

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1 Calulus TAYLO SEIES CONVEGENCE AND TAYLO EMAINDE Let the differee betwee f () ad its Taylor polyomial approimatio of order be (). f ( ) P ( ) + ( ) Cosider to be the remaider with the eat value ad the approimate value. is the error of the approimatio of by its Taylor poyomial of order. We eed to estimate. Taylor's Theorem whih is a geeralizatio of the Mea Value Theorem gives a formula for the remaider. Taylor's theorem If is differetiable through order i some ope iterval otaiig the for eah i there eists a umber betwee ad suh that ( ) f ( a) f ( ) f ( a) + a)( a) + L ( a) + ( )! where, the remaider, is give i the form ( + )! ( + ) + ( ) ( a) where lies betwee ad a. If for all i I we have lim ( ) 0 we write! the the Taylor Series of about overges to o ad If the absolute value of the derivative of is bouded above by a ostat the Taylor s Iequality provides a easier boud for Taylor s remaider. Taylor s Iequality: If all the oditios of Taylor's theorem are satisfied ad i additio there is a positive ostat suh that, for all betwee ad the Taylor s remaider satisfies the iequality K + ( ) ( a) ( + )! If these oditio hold for every the the series overges. (By the squeeze theorem. See below.) Fially reall we have a boud for the remaider of a alteratig series whih satisfies the oditios of the Alteratig Series Test. The truatio error, S S, has magitude less tha the magitude of the et term i the series. If the Taylor Series of about overges to i some iterval otaiig ad i additio the Taylor series of f () about a is a alteratig series for some i the iterval of overgee ad the alteratig series satisfies the oditios of the Alteratig Series Test the the truatio error is less tha the absolute value of the et term i the Taylor series ad the sig of the error is the sig of the et term. I this ase we kow the sig of the error whih Taylor's iequality does ot provide. Eample: Fid the iterval of overgee where the Taylor series for si about 0 overges to si. 7 The Taylor series for si about 0 is + + L whih overges absolutely for (, ). We!! 7! ow fid where it overges to si.the series has oly odd-powered terms ad for Taylor's theorem gives si!!! HdTaylorem.do Prof. L.A. Moth Page of

2 All the derivatives of si have absolute value less tha or equal to. By Taylor's Iequality k+ 0 k + ( ). (k + )! + k lim 0. By the Squeezig theorem lim k + ( ) 0 for every ad the MaLauri series for si k (k + )! k overges to si for every. We have just show the Taylor series for si about 0 overges to si o (, ) by showig that lim ( ) 0 o (, ). Similarly for os. Fially we a write 7 si + ( ) + + L (, )!! 7! 0 ( + )! 6 os + + L ( ) (, )!! 6! 0 ()! Note that the Taylor Series for si ad os are alteratig series for 0. Now let s eamie the Taylor Series for e. The MaLauri series for e, L,!! 0! overges for (, ). The Taylor Series for is a alteratig series for 0. By Taylor's theorem. e L +!!! ( + ) + + e ( ) where is betwee ad 0. ( + )! ( + )! lim ( ) e + lim is betwee ad 0. ( + )! Case. < 0, < < 0, e < 0 < ( ) e lim ( ) 0 lim ( ) < ( + )! ( + )! < 0 Case. > 0, 0 < <, 0 < ( ) e lim ( ) 0 lim ( ) 0 e < + e < e ( + )! > 0 + ( + )! We have just show the Taylor series of lim ( ) 0 for all. e about 0 overges to e o (, ) beause HdTaylorem.do Prof. L.A. Moth Page of

3 We ever eed to hek a beause the Taylor series of f () about a is a power series i a whih by defiitio f (a) at a. Eample: Fid Taylor's remaider for the Taylor series of with a ad. This meas fid the truatio error if we truate the Taylor series of about at the term of order. (The third Taylor polyomial) ( + )! ( + ) + ( ) ( a) betwee ad a ( + )!! 6 7 / ( ) ( ) betwee ad. 8 The alulatios for the derivatives of f () are show below. f () / ) / f ( ) / ( ) 8 () f ( ) 6 ( )! 8 (+ ) () + 7 / 7 / ( ) ( ) ( ) ( ) / 7 / Eample: Fid Taylor's remaider for the Taylor series of with a 0 ad. This meas fid the + truatio error if we truate the Taylor series of about 0 at the third Taylor polyomial. + ( + ) + ( ) ( a) betwee ad a ( + )! ( ) (+ ) ( + )! + f ( )! ()! ( + ) ( ) betwee ad 0. ( + ) The alulatios for the derivatives of f () are show below. f () ( + ) + ) ( + ) f ( ) ( + ) ( ) ( + ) ()! f ( )!( + ) ( + )! ( + ) HdTaylorem.do Prof. L.A. Moth Page of

4 Eample: Fid Taylor's remaider for the MaLauri series of ( + )! ( + )! ( ) ( + )! ( ) ( + ) ( ) ( ) betwee ad 0 + ( ) The alulatios for the derivatives of f () are show below. f () ( ) ) ( ) ( ) ( ) f ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ()! f ( )!( ) ( ) ( )! f ( ) + ( ) ( + ) ( + )! f ( ) + ( ) Eample: Prove + + L l l( + ) + L o (, ]. Note the "" sig. This is beause the HS is the Taylor series whih overges to the futio o (, ]. lim ( ) 0 i this iterval. To show + ( ) + + L + ( )( ) overges o (, ] use the ratio test. The series overges absolutely whe lim <. The radius ( + )( ) of overgee is. The iterval of overgee for is (, ). We eed to test the edpoits separately. At, + + ( ) ( ) whih overges oditioally. At, + + ( ) ( ) ( ) ( ) + + ( ) + + L whih diverges. + + l( ) L o (, ]. At, l HdTaylorem.do Prof. L.A. Moth Page of

5 Eample: Evaluate l usig a series whih overges faster tha eplae by i the Taylor series for l( + ) l( ) L o < l( ) L o < ( ) l( ) l L o [, ). At /, l + L For homework see HdTaylorem.do Prof. L.A. Moth Page of