TAYLOR SERIES, POWER SERIES


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1 TAYLOR SERIES, POWER SERIES The followig represets a (icomplete) collectio of thigs that we covered o the subject of Taylor series ad power series. Warig. Be prepared to prove ay of these thigs durig the exam. Thigs you should memorize: the formula of the Taylor series of a give fuctio f(x) geometric series (i.e. the Taylor expasio of x ) the Taylor expasios of the fuctios e x, si x, cos x, l( + x) ad rage of validity. the relatio f(x) = P (x) + R (x) ad Lagrage formula for R (x) You should also uderstad the actual proofs of the Taylor series expasios eumerated above.. TAYLOR SERIES f(x) = f () () x R (x) I other words, the Taylor expasio takes place oly at those values of x for which R (x). If you wat to prove from scratch a Taylor series expasio (as we did i the case of e x, cos(x), si(x) ad l( + x)) you eed to show R (x), ad oe usually proves this by employig Lagrage formula estimatig R (x) (get rid of c) See the slides of Nov 24 lecture. Expoetial fuctio. e x = x, x R Uderstad why this gives, amog others, the followig formula 2 = ( ) e, = e Cosie. Kow how to estimate the remaider i this case to prove cos x = ( ) x2 (2)!, x R I particular this gives (set x = /2) ( ) 4 = cos.5 (2)! Sie. si x = ( ) x2+ (2 + )!, x R
2 2 TAYLOR SERIES, POWER SERIES Hyperbolic Cosie. From oe derives e x x = ad e x = ( ) x e x + e x x 2 = 2 (2)!, If we set x = we obtai form example (2)! = e + /e 2 as opposed to Geometric series. = e ad ( ) (2)! = cos() x R x = x holds oly for < x < Logarithm. Start with the fake geometric series ( ) x = + x Itegrate (apply the ice theorem o power series): l( + x) = ( ) x+ + = ( ) + x, x (, ) If we wat to justify this idetity i the rage S = (, ], we eed to appeal to Abel s theorem. I particular, for x = we get ( ) + l(2) = = Equivaletly, we have the followig power expasio i x ( ) + l(x) = (x ) valid for < x 2 Approximate Computatios. Startig with f(x) = P (x) + R (x) for a give f(x), oe ca presumably fid such that R (x) is smaller tha the desired degree of accuracy (estimate R (x)!) i order to kow that P (x) approximates f(x) well eough. Examples: computig e.2, /e, / e, si.5 to three decimal places (i.e. approximate the fuctio by a appropriate Taylor polyomial, etc.)
3 TAYLOR SERIES, POWER SERIES 3 Example ot doe i class: compute l(.4) to 2 decimal places by approximatig the fuctio l( + x) by Taylor polyomial. 2. POWER SERIES Give a power series a x, oe ca determie: The radius of covergece R with the formula R = lim a / The domai of covergece S which cosists of all the umbers x for which the series a x is coverget: the ope iterval ( R, R) is for sure icluded, ad the we oly have to check the edpoits x = ±R separately. The power series is diverget outside this rage, i.e. for x > R. Example. Fid the radius of covergece R ad the domai of covergece S for each of the followig power series: x x,, x, x x,, ( ) 2 x 2 Hwk problem: if the series k= 4 a is coverget, the a ( 2) is also coverget. (the questio reduces to uderstadig the shape of the domai of covergece S of the power series a x ) 2.. The "Nice Theorem". The ice theorem allows us to differetiate/itegrate a Taylor series expasio iside the radius of covergece, i order to obtai ew idetities (Taylor series expasios). If f(x) = a x, x ( R, R) the f (x) = a x f(t)dt = a + x+ (differetiate) (itegrate) The coefficiets of the power series obtaied through differetiatio are a. The coefficiets of the power series obtaied through itegratio are a +. The above two idetities are valid wheever x ( R, R) Side Remark. Why is this thig called a theorem? To give a simple example, let It is easy to differetiate ad itegrate g(x): g(x) = 2x + x 3 + x 4 g (x) = 2 + 3x 2 + 4x 3 g(t) = x 2 + x4 4 + x5 5 Now, the ice theorem says that i ca do the same thig eve if g(x) was ot a polyomial (fiite sum of powers), but a power series (ifiite sum of powers!). However whe dealig with a power series we are facig the issue of covergece, ad the process of
4 4 TAYLOR SERIES, POWER SERIES differetiatio (itegratio) termbyterm eeds justificatio. The ice theorem takes care of that. Also, thik of the ice theorem as allowig as to obtai ew idetities from old oes. Example. Start with x = x, x (, ). Differetiate/itegrate: x = ( x) 2 (differetiatio) x = dx = l( x) (itegratio) x ad these idetities are valid for x (, ). We ca multiply both sides of the first oe to obtai x = x ( x), x 2 (, ). For example, takig x = 4 gives ( ) 4 = We ca take x = i the secod idetity (Abel s theorem) to obtai ( ) Example 2. Start with Itegrate: = l(2). I other words, ( ) + + x 2 = ( ) x 2, x (, ) ta x = ( ) x2+ Exted this idetity to x = (ok by Abel s theorem):, x (, ) 2 + π 4 = ( ) 2 + = = l(2) Example 3. Put x 2 i the Taylor expasio of the expoetial fuctio to obtai the idetity e x2 = ( ) x2, x R Itegrate: For x = we get e t2 dt = ( ) x 2+ (2 + ), e t2 dt = ( ) (2 + ) x R This allows us to compute the itegral o the lefthad side (otherwise hard to figure out) to desired accuracy, as i the Example 7 o page 695 of the textbook. Example 4. Fid the sum of the series ( + 2) Start with the power series expasio e x = x
5 TAYLOR SERIES, POWER SERIES 5 Multiply both sides by x Itegrate xe x = te t dt = x + Evaluate itegral o the left by itegratio by parts Therefore Set x = to obtai Set x = 2 to obtai x +2 ( + 2) te t dt = te t x e t = xe x e x + xe x e x + = ( + 2) = x +2 ( + 2) 2 ( + 2) = e RELATION BETWEEN TAYLOR SERIES AND POWER SERIES A power series = Taylor series of its sum I other words, every time you obtai a idetity a x = (somethig) the the power series o the lefthad side must be the Taylor series of that somethig o the righthad side. Example. We kow that e x = holds for ay x R (we proved this statemet by meas of the remaider formula). Therefore there is o harm i cosiderig x 2 istead of x i the above "formula", oly to obtai x 2 e x2 = Lookig at the boxed priciple (above), we ca ow see that what we have i fact here is the Taylor expasio of the fuctio e x2 which we obtaied almost for free. (Covice yourselves that it is ot so trivial to costruct the Taylor series of the fuctio f(x) = e x2 from scratch. Not to metio that to justify the Taylor series expasio oe usually eeds to show that R (x), ad i the case of f(x) = e x2 Lagrage s formula for the remaider is really complicated. x
6 6 TAYLOR SERIES, POWER SERIES Example 2. A simpler example is the idetity + x 2 = ( ) x 2 which is valid for x (, ) ad obtaied from the geometric series (simply by replacig x by x 2 ). I view of the boxed priciple above, this has to be the Taylor expasio of the fuctio g(x) = +x. Hece, if you eed to compute g () () simply idetify the 2 coefficiet of x : g () () x = ( ) x 2 (for some ) 2 =, = 5! Therefore g () ()! = ( ) 5 = g () () =!
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