Power Series Expansions of Binomials
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1 Power Series Expasios of Biomials S F Ellermeyer April 0, 008 We are familiar with expadig biomials such as the followig: ( + x) = + x + x ( + x) = + x + x + x ( + x) 4 = + 4x + 6x + 4x + x 4 ( + x) 5 = + 5x + 0x + 0x + 5x 4 + x 5 The above expaded polyomials ca be thought of as power series with itely may terms For example, the expasio of ( + x) 4 is actually the MacClauri Series for ( + x) 4 It ca be obtaied by elemetary algebraic calculatio (repeated use of the distributive property) or it ca be obtaied i the usual way of calculatig the coe ciets of a MacClauri Series: For example, for f (x) = ( + x) 4, we have f (0) (x) = ( + x) 4 f () (x) = 4 ( + x) f () (x) = 4 ( + x) f () (x) = 4 ( + x) f (4) (x) = 4 f (5) (x) = 0 f (6) (x) = 0
2 ad f (0) (0) = ( + 0) 4 = f () (0) = 4 ( + 0) = 4 f () (0) = 4 ( + 0) = f () (0) = 4 ( + 0) = 4 f (4) (0) = 4 = 4 f (5) (0) = 0 f (6) (0) = 0 which gives a 0 = 0! = a = 4! = 4 a =! = 6 a = 4! = 4 a 4 = 4 4! = a 5 = 0 a 6 = 0 Note that a = 0 for all 5 Thus the MacClauri series of f (x) = ( + x) 4 is + 4x + 6x + 4x + x 4 By examiig the above calculatios of the umbers a, we see that this series ca also be writte as 0! + 4! x + 4 x + 4 x + 4 x 4!! 4! or as 4! 0!4! + 4!!! x + 4!!! x + 4!!! x + 4! 4!0! x4
3 Sice the above series cotais oly itely may o zero terms, there is o problem i determiig the radius of covergece The radius of covergece is It is also true (ad ca be proved by purely algebraic meas) that the fuctio f (x) = ( + x) 4 is equal to its MacClauri Series I other words, the equatio ( + x) 4 = 4! 0!4! + 4!!! x + 4!!! x + 4!!! x + 4! 4!0! x4 is true for all real umbers, x I geeral, if is ay positive iteger, the for all real umbers, x, it is true that where ( + x) = = X =0 x!! ( )! The umbers are called biomial coe ciets For each positive iteger, the umbers ; 0, mae up the th row of Pascal s Triagle: Exercises Compute the followig biomial coe ciets (a) 5 (b)
4 7 (c) 0 6 (d) 6 (e) Obtai the biomial expasio (MacClauri Series) of (a) ( + x) 6 (b) ( + x) 7 (c) ( x) (a) (Experimetatio) Try addig all of the biomial coe ciets that correspod to the same value of For example compute , 0 4 etc What patter do you otice i these sums? (b) (Veri catio) We ow that the equatio ( + x) = X =0 x is true for all real umbers x Verify that your cojecture from part a is correct by pluggig a appropriate value of x ito this equatio 4 (a) (Experimetatio) Pic some positive iteger ad some iteger such that ad compute + 4
5 Try this for several di eret values of What patter do you observe Cojecture a formula: + =? (b) (Veri catio) Use the fact that =!! ( )! to prove (algebraically) that your cojecture is correct 5 Let a ad b be ay two real umbers (costats) ad let be a positive iteger (a) Write the biomial expasio for + b a (b) By observig that (a + b) = a + b = a + b, a a ad by usig your result from part a, write the biomial expasio of (a + b) Power Series Expasios of ( + x) Where is a Negative Iteger Let us cosider the problem of dig the MacClauri Series for This ca be doe by recallig that f (x) = ( + x) x = X x = + x + x + x + x 4 =0 for all x such that jxj < ad the by maig a substitutio of of x: =0 x i place ( + x) = + x = ( x) = X ( x) = x + x x + x 4 5
6 The above series expasio also holds for all x such that jxj < Aother way (the direct way) to d the MacClauri Series for f (x) = ( + x) is as follows: f (0) (x) = ( + x) f () (x) = ( ) ( + x) f () (x) = ( ) ( ) ( + x) f () (x) = ( ) ( ) ( ) ( + x) 4 f (4) (x) = ( ) ( ) ( ) ( 4) ( + x) 5 f (5) (x) = ( ) ( ) ( ) ( 4) ( 5) ( + x) 6 ad f (0) (0) = ( + 0) = f () (0) = ( ) ( + 0) = f () (0) = ( ) ( ) ( + 0) = f () (0) = ( ) ( ) ( ) ( + 0) 4 =! f (4) (0) = ( ) ( ) ( ) ( 4) ( + 0) 5 = 4! f (5) (0) = ( ) ( ) ( ) ( 4) ( 5) ( + 0) 6 = 5! 6
7 which gives a 0 = 0! = a = =! a =!! = a =! =! a 4 = 4! 4! = a 5 = 5! 5! = Thus the MacClauri series of f (x) = ( + x) is x + x x + x 4 By examiig the above calculatios of the umbers a, we see that this series ca also be writte as 0! +! ( ) ( ) x + x +! ( ) ( ) ( ) x +! ( ) ( ) ( ) ( 4) x 4 + 4! I geeral, if is ay egative iteger, the for all umbers, x, such that jxj <, it is true that where ( + x) = 0 X =0 = x ad ( ) ( ) ( ( )) =! The umbers are called geeralized biomial coe ciets 7
8 Exercises Compute the followig geeralized biomial coe ciets (a) (b) (c) 0 (d) 5 7 (e) Obtai the biomial expasio (MacClauri Series) of (a) ( + x) (b) ( + x) (c) ( x) Geeral Biomial Expasios The results give that we have discovered for the biomial expasios of ( + x) where is a positive or egative iteger geeralize completely to the case that ca be ay real umber For ay real umber ad ay o egative iteger we de e the geeralized biomial coe ciets by = 0 ad ad it is true that = ( ) ( ) ( ( ))! ( + x) = X =0 8 x
9 for all x such that jxj < This result is called the Biomial Theorem It is ot di cult to derive the biomial (MacClauri) series for ay fuctio of the form ( + x) ad to show that ay such series has radius of covergece R = The oly di culty is to prove that this MacClauri Series is i fact equal to ( + x) for all x such that jxj < A clever proof of the latter fact is give as a exercise i Stewart s textboo Exercises Compute the followig geeralized biomial coe ciets (a) (b) (c) (d) 5 (e) 4 Obtai the biomial expasio (MacClauri Series) of (a) ( + x) = (b) ( + x) =4 (c) ( x) = (d) ( + x ) = Use the reasoig outlied i Exercise 5 i Sectio 88 of the textboo to prove that X ( + x) = x for all x such that jxj < =0 9
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