Computing Taylor Series Lecture Notes

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Dwayne Kennedy
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1 Computing Tylor Series Lecture Notes As e hve seen, mny different functions cn e expressed s poer series. Hoever, e do not yet hve n explntion for some of our series (e.g. the series for /, sin, nd cos), nd e do not hve generl formul for finding Tylor series. In this section e ill lern ho to find Tylor series for virtully ny function. The Tylor Series Formul A generl poer series cn e expressed s $ % 0 œ -! - - -$ -% â, here -! ß- ß- ßá re constnts. As ith polynomil, e often don't other to rite terms tht hve coefficient of!, ut e cn imgine tht every poer series hs every one of these terms. The first term of poer series is clled the constnt term. The constnt term is ht you get hen you sustitute in œ!. For exmple, if then the constnt term of $ % 0 œ $ & ( * â, 0 is $, so 0! œ $!!!! â œ $. The second term of poer series is clled the liner term or term, nd hs the form - for some coefficient -. You cn otin the coefficient - y tking the derivtive of the series nd then sustituting œ!. For instnce, if then $ % 0 œ $ & ( * â $ 0 œ & % ( %% â, so 0! œ &. As you cn see, the coefficient of in 0 is the sme s the constnt term of 0, nd is therefore equl to 0!. In generl, tking the derivtive of poer series demotes ech of the coefficients y one step:

2 $ 0 œ $ & ( * â Æ Æ Æ â 0 œ & % ( â The coefficient of ecomes the constnt term, the coefficient of ecomes the coefficient $ of (nd is multiplied y ), the coefficient of ecomes the coefficient of (nd is multiplied y $ ), nd so forth. The folloing formul reltes the coefficients of poer series to the vlues of the derivtives t!: FORMULA FOR THE COEFFICIENTS _ Let 0 œ - œ! e poer series. Then: - œ 0! x here 0 denotes the th derivtive of 0. The folloing clcultion illustrtes this pttern: $ % &! $ % & If 0 œ â, $ % $ % & then 0 œ - - $- %- &- â, $ $ % & 0 œ - '- -!- â, $ 0 œ '- %- '!- â, % $ % 0 œ %-!- â, & 0 œ!- â, & % & & ã ä As you cn see, the constnt term of 0 is lys equl to x multiplied y - : 0! œ x This explins the formul for the coefficients given ove. The formul ove cn e used to find Tylor series for virtully ny function. In generl, function is clled nlytic if it cn someho e represented y poer series. Most functions defined y formul re nlytic, nd e no kno ho to find the Tylor series for ny nlytic function:

3 TAYLOR SERIES FORMULA Let 0 e ny function, nd suppose tht 0 is nlytic. Then _ 0! 0 œ x œ! $ 0! 0! $ 0! œ 0! 0! â â. x $x x EXAMPLE 1 Assuming tht / is nlytic, find the Tylor series for /. SOLUTION Let 0 œ /. Then 0 œ /, 0 œ /, nd so on, so 0! œ ß 0! œ ß 0! œ ß á We conclude tht $ % / œ â x $x %x EXAMPLE 2 Assuming tht sin is nlytic, find the Tylor series for sin. SOLUTION Let 0 œ sin. Here re the first seven derivtives: 0 œ sin so 0! œ sin! œ! 0 œ cos so 0! œ cos! œ 0 œ sin so 0! œ sin! œ! $ $ 0 œ cos so 0! œ cos! œ % % 0 œ sin so 0! œ sin! œ! & & 0 œ cos so 0! œ cos! œ ' ' 0 œ sin so 0! œ sin! œ! ( ( 0 œ cos so 0! œ cos! œ This pttern ill continue to repet. Therefore, the Tylor series for sin is:! $! % &! ' ( sin œ! â x $x %x &x 'x (x $ & ( œ â $x &x (x

4 Don't forget tht there re other ys to find the Tylor series for function. You only need to use the formul if no other method is ville: EXAMPLE 3 Find the Tylor series for tn. SOLUTION There is no need to use the Tylor series formul here. We cn otin poer series for tn y plugging into the Tylor series for tn : tn ˆ œ â $ & ( '! % EXAMPLE 4 Find the Tylor series for 0 œ. SOLUTION: 0 œ so 0! œ $ 0 œ so 0! œ % 0 œ ' so 0! œ ' $ & $ 0 œ % so 0! œ % % ' 0 œ! so 0 %! œ! Therefore, ' % $! % œ â x $x %x $ % œ $ % & â In the folloing exmple, it is someht complicted to find pttern in the coefficients, mking it difficult to find more thn the first fe terms: EXAMPLE 5 Find the first three terms of the Tylor series for 0 œ È. SOLUTION 0 œ so 0! œ Î 0 œ Î so 0! œ 0 œ $Î so 0! œ % %

5 Therefore, the first three terms of the Tylor series for È re: 0! Î% 0! 0! œ x œ ) e re tht mny functions still cnnot e expressed s poer series using this formul. For exmple, the function 0 œ Î hs no Tylor series, since 0! is undefined. In generl, ny function for hich 0! is undefined for some ill fil to e nlytic. Generl Tylor Series So fr, e hve only een deling ith poer series centered t œ!: _ - œ â œ!! Such series tends to converge hen is close to!, nd diverge hen is fr y from!. A more generl form for poer series is: _ 0 œ - + œ â œ!! This is clled poer series centered t œ +. The dvntge of this series is tht it tends to converge hen is close to +. For poer series centered t œ +, the formul for the th coefficient is - œ 0 + x GENERAL TAYLOR SERIES Let 0 e function, nd suppose tht 0 is nlytic t œ +. Then: _ œ + x œ! œ $ + $ + â. x $x The formul ove uses the phrse nlytic t œ +, hich mens tht 0 cn e expressed s poer series centered t œ +.

6 EXAMPLE 6 Find the Tylor series for 0 œ Î centered t œ. SOLUTION We hve: 0 œ Î so 0 œ $ 0 œ so 0 œ % 0 œ ' so 0 œ ' $ & $ 0 œ % so 0 œ % % ' % 0 œ! so 0 œ! Therefore, ' % $! % œ â x $x %x $ % œ $ % & â It is lso possile to otin Tylor series centered t œ + using sustitution. For exmple, e kno the formul $ % & ln œ â $ % & Plugging in for gives the Tylor series for ln centered t œ : ln œ $ % & â $ % & Note tht ln does not hve Tylor series centered t œ!, since ln! is undefined.

7 EXERCISES $ 1. Let 0 œ /. $ () Find 0!, 0!, 0!, nd 0!. () Wht is the generl formul for 0!? (c) Use your nser from prt () to find the Tylor series for / $. 2. Let 0 œ. $ % & () Find 0!, 0!, 0!, 0!, 0!, nd 0!. () Wht is the generl formul for 0!? (c) Use your nser from prt () to find the Tylor series for. 3. Find the Tylor series for 0 œ. Express your $ nser using summtion nottion. ' 4. Find the Tylor series for 0 œ. Express your % nser using summtion nottion. 5. Find the first four terms of the Tylor series for È $. 6. Find the first four terms of the Tylor series for È %. 7. Use the Tylor series formul to find the Tylor series for cos ç Find the Tylor series for 0 ithout using the Tylor series formul. Express your nser using summtion nottion. & 9. 0 œ / œ œ ln / œ sin $ > œ sin œ ( /.> 15 1 ç Find the first three terms of the Tylor series for 0 centered t the given vlue of œ È, + œ & œ È $, + œ ) œ tn, + œ 1. 0 œ tnß + œ % ç Find the Tylor series for 0 centered t the given vlue of œ /, + œ $ œ /, + œ & œ sin, + œ œ cos, + œ 1 % $ œ, + œ œ &, + œ & œ, + œ œ, + œ (!. Use the Tylor series formul to find the Tylor series for ln.

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered: Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you