Mathematical Methods for Physics and Engineering - Matematica.NET

PRELIMINARY ALGEBRAIn fact, the general expression, the binomial expansion for power n, is given by∑k=n(x + y) n =n C k x n−k y k , (1.49)k=0where n C k is called the binomial coefficient and is expressed in terms of factorialfunctions by n!/[k!(n − k)!]. Clearly, simply to make such a statement does notconstitute proof of its validity, but, as we will see in subsection 1.5.2, (1.49) canbe proved using a method called induction. Before turning to that proof, weinvestigate some of the elementary properties of the binomial coefficients.1.5.1 Binomial coefficientsAs stated above, the binomial coefficients are defined by( )n n! nC k ≡k!(n − k)! ≡ for 0 ≤ k ≤ n, (1.50)kwhere in the second identity we give a common alternative notation for n C k .Obvious properties include(i) n C 0 = n C n =1,(ii) n C 1 = n C n−1 = n,(iii) n C k = n C n−k .We note that, for any given n, the largest coefficient in the binomial expansion isthe middle one (k = n/2) if n is even; the middle two coefficients (k = 1 2(n ± 1))are equal largest if n is odd. Somewhat less obvious is the resultn C k + n n!C k−1 =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+1 C k . (1.51)An equivalent statement, in which k has been redefined as k +1,isn C k + n C k+1 = n+1 C k+1 . (1.52)1.5.2 Proof of the binomial expansionWe are now in a position to prove the binomial expansion (1.49). In doing so, weintroduce the reader to a procedure applicable to certain types of problems andknown as the method of induction. The method is discussed much more fully insubsection 1.7.1.26