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) term-by-term 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 left-had 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 left-had side must be the Taylor series of that somethig o the right-had 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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