FIRST YEAR CALCULUS W W L CHEN

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1 FIRST YEAR CALCULUS W W L CHEN c W W L Che, 994, This chapte is available fee to all idividuals, o the udestadig that it is ot to be used fo fiacial gai, ad may be dowloaded ad/o photocopied, with o without pemissio fom the autho. Howeve, this documet may ot be kept o ay ifomatio stoage ad etieval system without pemissio fom the autho, uless such system is ot accessible to ay idividuals othe tha its owes. Chapte 22 THE BINOMIAL THEOREM 22.. Fiite Biomial Expasios I may istaces, oe eeds to study expessios like (a + b), whee N {0}. Let us fist of all look at a few small values of. It is ot difficult to see that (a + b) 0, (a + b) a + b, (a + b) 2 a 2 + 2ab + b 2, (a + b) 3 a 3 + 3a 2 b + 3ab 2 + b 3, (a + b) 4 a 4 + 4a 3 b + 6a 2 b 2 + 4ab 3 + b 4, ad so o. We ca display the coefficiets i the fom of the Pascal tiagle below: () Of couse, thee is o easo to stop at 4. If we go o idefiitely, the fo each N {0}, we ca wite (a + b) c,0 a + c, a b c, ab + c, b, Chapte 22 : The Biomial Theoem page of 5

2 Fist Yea Calculus c W W L Che, 994, 2008 whee the coefficiets give ise to the ow c,0 c,... c, c, i the Pascal tiagle. Howeve, what ae the values of these coefficiets? To fid the values of these coefficiets, we fist make two obsevatios. () I the Pascal tiagle (), evey ety is the sum of the two eties immediately above it. Fo example, highlights the fact c 3,0 + c 3, c 4,. So is it tue that wheeve N ad? c, + c, c +, (2) Evey ow i the Pascal tiagle () stats ad eds with the ety. So is it tue that wheeve N {0}? c,0 c, The aswe to these two questios ae give by the followig esult. PROPOSITION 22A. (BINOMIAL THEOREM) Fo evey N {0}, we have whee, fo evey 0,,...,, we have (a + b) c,0 a + c, a b c, ab + c, b, (2) c, ( )... ( + ), (3)! with the covetio that 0! ad that the expessio ( )... ( + ) epesets whe 0. Poof. We shall pove this esult by iductio o. Suppose that fo evey N {0} ad evey 0,,...,, the tem c, is give by (3). Note that c 0,0 c,0 c,, so that (2) holds whe 0 ad. Suppose ow that fo fixed, we have The (a + b) c,0 a + c, a b c, ab + c, b. (a + b) + (a + b)(a + b) (a + b)(c,0 a + c, a b c, ab + c, b ) (c,0 a + + c, a b c, a 2 b + c, ab ) + (c,0 a b + c, a b c, ab + c, b + ) c,0 a + + (c,0 + c, )a b + (c, + c,2 )a b (c, + c, )ab + c, b +. (4) Chapte 22 : The Biomial Theoem page 2 of 5

3 Fist Yea Calculus c W W L Che, 994, 2008 Note ow that O the othe had, if, the Combiig (4) (6), we coclude that c,0 c +,0 ad c, c +,+. (5) ( )... ( + 2) ( )... ( + ) c, + c, + ( )!! ( ( )... ( + 2) + + ) ( )! ( )... ( + 2) + ( )! ( + )( )... ( + + ) c +,. (6)! (a + b) + c +,0 a + + c +, a b c +, ab + c +,+ b +. The esult ow follows fom the Piciple of iductio. Remak. We usually wite ( )... ( + ),! so that the Biomial theoem becomes (a + b) a + 0 a b ab + b. I fact, the biomial coefficiet (7) epesets the umbe of ways of choosig objects fom a collectio of objects. The easo that (7) is the coefficiet fo a b i the expasio of (a + b) is as follows: Sice (a + b) (a + b)... (a + b), }{{} it follows that fom these copies of (a + b), we eed to pick a exactly ( ) times ad pick b exactly times ad multiply i ode to get a tem a b. It follows that the coefficiet fo a b is the umbe of diffeet ways that we ca pick a exactly ( ) times ad pick b exactly times, ad this is the biomial coefficiet (7). Remaks. () Fo evey N ad evey,...,, we have + ( + ). (2) Lettig a b i the Biomial theoem, we see that fo evey N {0}, we have Chapte 22 : The Biomial Theoem page 3 of 5

4 Fist Yea Calculus c W W L Che, 994, Ifiite Biomial Expasios Sometimes, we eed to study expessios like ( + x) α, whee α R is ot a o-egative itege. We ca wite dow a seies expessio fo the fuctio as follows. Howeve, we eed to be caeful about covegece of the seies. PROPOSITION 22B. (EXTENDED BINOMIAL THEOREM) Suppose that α R. The fo evey x R satisfyig x <, we have ( + x) α 0 α x, whee fo evey 0,, 2,..., the exteded biomial coefficiet is give by α α(α )... (α + ).! Popositio 22B ca be demostated usig Taylo seies; see Sectio 2.2. Note also that Popositio 22B educes to the Biomial theoem fo a ad b x whe α is a o-egative itege. The special case whe α is a egative itege is somewhat special, as we ca calculate the exteded biomial coefficiets athe easily. PROPOSITION 22C. Suppose that m N. The fo evey 0,, 2,..., we have m m + ( ). Poof. We have m as equied. m( m )... ( m + )! m(m + )... (m + ) ( )! ( (m + )(m + 2)... (m + + ) m + ( ) ( )! ) Chapte 22 : The Biomial Theoem page 4 of 5

5 Fist Yea Calculus c W W L Che, 994, 2008 Poblems fo Chapte 22. Fid the fist fou tems of the seies fo ( + 2x) What is the coefficiet of x 4 i the seies expasio of ( + x + x 2 ) 4? Chapte 22 : The Biomial Theoem page 5 of 5