Taylor and Maclaurin series, Part 2.

Transcription

1 MATH A1 - Sprig Taylor ad Maclauri series, Part 2. Recall the Biomial Theorem says if k is a positive iteger, we may expad (1+x) k usig the formula k (1 + x) k = x Here ( k ) deote the biomial coefficiets: :=, := 1. 0 { Note: Sice here 0 k are itegers, i fact the biomial coefficiets take the form k = k!. I this cotext ( k (k )! ), read k choose, is the umber of ways to select objects from a set cotaiig k objects. } Example 1. (a) Fid the Maclauri series for (1 + x) k, where k is ay real umber. (b) Show that this series represets (1 + x) k for x < 1. Solutio. (a) Arragig our work i colums, we have f(x) = (1 + x) k f(0) = 1 f (x) = k (1 + x) k 1 f (0) = k f (x) = k(k 1) (1 + x) k 2 f (0) = k(k 1) f (x) = k(k 1)(k 2) (1 + x) k 3 f (0) = k(k 1)(k 2).. f () (x) = k(k 1) (k + 1) (1 + x) k f () (0) = k(k 1) (k + 1) Therefore the Maclauri series of f(x) = (1 + x) k is k = lim + 1 f () (0) x = x = lim x. Notice the similarity betwee the coefficiets of this series ad the biomial coefficiets. The differece is that ow k is allowed to be ay real umber. This series is called the biomial series. To fid its radius of covergece, let a deote the -th term. We have L = lim a +1 a = lim (k )x +1 ( + 1)! x k x = x.

2 MATH A1 - Sprig The, by the ratio test, the series coverges if x < 1 ad diverges if x > 1; so the radius of covergece is R = 1. (b) Let s defie g(x) = 1 + x. We wat to show that g(x) = (1 + x) k for x < 1. We kow that we ca differetiate g(x) term-by-term withi its radius of covergece: m=0 g (x) = x 1. (*) ( 1)! Chagig the idex of summatio by m = 1 we ca write g k(k 1)(k 2) (k m) (x) = x m k(k 1)(k 2) (k ) = x. (**) m! Addig g (x) from (**) to x times g (x) from (*), we get (1 + x)g k(k 1)(k 2) (k ) (x) = x + = k + = k + k x [(k ) + ] x = kf(x). x ( 1)! The equatio (1 + x)g (x) = kg(x), which holds for x < 1, implies that d g(x) = (1 + x)k g (x) k(1 + x) k 1 g(x) = 0. dx (1 + x) k (1 + x) 2k Thus g(x) (1+x) k is costat o the iterval ( 1, 1), ad sice its value at x = 0 is 1, the costat must be 1. We coclude that g(x) = (1 + x) k for x < 1, which shows that the fuctio (1 + x) k is equal to the sum of its Maclauri series. Remark. We exted the defiitio of biomial coefficiets settig, for ay real umber k ad itegers 0, k k :=, := 1. 0 This way we ca write the result of the example i a way esay to remember: The Biomial Series. If k is ay real umber, the (1 + x) k = x, for x < 1.

3 MATH A1 - Sprig Approximatig Fuctios by Polyomials. Suppose that f(x) is equal to the sum of its Taylor series at the poit a: f(x) = Recall from Taylor s formula that f () (a) (x a), for x a < R. f(x) = T (x) + R (x); where withi the radius of covergece R > 0 of the Taylor series, the remaider coverges to zero: lim R (x) = 0, for x a < R. Put it this way: f(x) = lim T (x), for x a < R. This meas that withi the radius of covergece of the Taylor series, we ca use the Taylor polyomials T (x) to approximate values of the fuctio f(x). Furthermore, we ca cotrol the size of the error i this approzimatio as follows:. (a) If the Taylor series is a alteratig series, R (x) (x a) +1 (b) I all cases, we ca use Taylor s iequality. f (+1) (a) (+1)! Example 2. (This is Example 1 from Sectio of the textbook.) (a) Approximate the fuctio f(x) = 3 x by a Taylor polyomial of degree 2 at a = 8. (b) How accurate is this approximatio whe 7 x 9? Example 3. (This is Example 2 from Sectio of the textbook.) (a) What is the maximum error possible i usig the approximatio si x x x3 3! + x5 5! whe 0.3 x 0.3? (b) Use this approximatio to fid si 12 correct to six decimal places. (c) For what values of x is this approximatio accurate to withi ? Exercise: Read page 745 of the texbook, about multiplicatio ad divisio of power series.

4 MATH A1 - Sprig Solutio of Example 2. Arragig our work i colums, f(x) = 3 x = x 1/3 f(8) = 2 f (x) = 1 3 x 2/3 f (8) = 1 12 f (x) = 2 9 x 5/3 f (8) = f (x) = x 8/3 Thus the secod degree Taylor polyomial is T 2 x = f(8) + f (8) (x 8) + f (8) (x 8) 2 1! 2! = (x 8) (x 8)2. So usig this polyomial to approximate 3 x meas that we may write 3 x T2 (x) = (x 8) (x 8)2. (b) The Taylor series is ot a alteratig series whe x < 8, so we ca t use the Alteratig Series Estimatio Theorem i this example. But we ca use Taylor s iequality with = 2 ad a = 8. To this ed, we must fid a positive umber M such that f (x) M wheever 7 x 9, or equivaletly, wheever x 8 1. Notice that x 7 iplies x 8/3 7 8/3, ad hece f (x) = x8/ < /3 Therefore we ca take M = The Taylor s iequality gives R 2 (x) ! x ! 1 3 = < This says that if 7 x 9, the approximatio i part (a) is accurate to withi Solutio of Example 3 (from pages 736, 738 of the textbook). (a) Notice that the Maclauri series si x = ( 1) x 2+1 (2 + 1)! = x x3 3! + x5 5! x7 7! + is a alteratig series for all x 0, ad the successive terms decrease i size because x 0.3 < 1. The we ca use the Alteratig Series Estimatio Theorem: The absolute value of the remaider is domiated by the absolute value of the first eglected term. Thus, the error i approximatig si x by the first three terms of its Maclauri series is at most x 7 7! = x

5 MATH A1 - Sprig Sice we are lookig at values of x satisfyig x 0.3, the error is at most (0.3) (b) To fid si 12 we first covert the agle to radias: 12π ( π si 12 = si = si ) π ( π ) ( π ) ! ! Thus, correct to six decimal places, si (c) The error will be withi if x < , or equivaletly x 7 < So the give approximatio is accurate to withi whe x < 0.82.