Mathematical Methods for Physics and Engineering : A Comprehensive Guide

(Darren Dugan) #1

PRELIMINARY ALGEBRA


In fact, the general expression, thebinomial expansionfor powern, is given by

(x+y)n=

∑k=n

k=0

nC
kx

n−kyk, (1.49)

wherenCkis called thebinomial coefficientand is expressed in terms of factorial


functions byn!/[k!(n−k)!]. Clearly, simply to make such a statement does not


constitute proof of its validity, but, as we will see in subsection 1.5.2, (1.49) can


beprovedusing a method called induction. Before turning to that proof, we


investigate some of the elementary properties of the binomial coefficients.


1.5.1 Binomial coefficients

As stated above, the binomial coefficients are defined by


nC
k≡

n!
k!(n−k)!


(
n
k

)
for 0≤k≤n, (1.50)

where in the second identity we give a common alternative notation fornCk.


Obvious properties include


(i)nC 0 =nCn=1,
(ii)nC 1 =nCn− 1 =n,
(iii)nCk=nCn−k.

We note that, for any givenn, the largest coefficient in the binomial expansion is


the middle one (k=n/2) ifnis even; the middle two coefficients (k=^12 (n±1))


are equal largest ifnis odd. Somewhat less obvious is the result


nC
k+

nC
k− 1 =

n!
k!(n−k)!

+

n!
(k−1)!(n−k+1)!

=

n![(n+1−k)+k]
k!(n+1−k)!

=

(n+1)!
k!(n+1−k)!

=n+1Ck. (1.51)

An equivalent statement, in whichkhas been redefined ask+1, is


nC
k+

nC
k+1=

n+1C
k+1. (1.52)

1.5.2 Proof of the binomial expansion

We are now in a position toprovethe binomial expansion (1.49). In doing so, we


introduce the reader to a procedure applicable to certain types of problems and


known as themethod of induction. The method is discussed much more fully in


subsection 1.7.1.

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