Higher Algebra

CHAPTER XXXIV.

THE BINOMIAL THEOREM.

§ 1. THE BINOMIAL THEOREM FOR POSITIVE INTEGRAL

EXPONENTS.

1. In Ch. XXVIII., it was proved by induction that, when n is a positive integer,

We will here give a briefer proof, based upon the theory of combinations.

Consider the following continued product of n factors:

a + b a + b

n factors

a + b

The first term of the product is formed by taking an a from each factor, giving a". The second term is formed by taking an a from n - 1 factors and a b from the remaining factor, giving a"-1b. But such a term can be formed in as many ways as one b can be taken from n b's, i.e., in C1 ways. Therefore the product so far is a" + Cian-1b.

A third term is formed by taking an a from n - 2 factors and a b from the remaining two factors, giving a"-262. But such a term can be formed in as many ways as two b's can be taken from n b's, i.e., in „C2 ways. Consequently, the product to this point is a" + „C1a”-1b + „C2an-262. In general, an a can be taken from each of n k+1 factors and a b from each of the remaining k-1 factors, giving an-*+1b-1. But such a term can evidently be formed in C-1 ways. We thus obtain

(a + b)" = an + „C1an-1b + „C2an-2b2 +

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Properties of Binomial Coefficients.

2. The kth term, counting from the beginning of the expansion, contains b-1, and is C--+-1. The kth term, counting from the end, contains a*-, and therefore b-*+, and is nC'n-k+1α-b"-k+1.

But, by Ch. XXX., § 3, Art. 3, „Ck-1 = „Cn-k+1 We therefore conclude:

In the expansion of (a+b)", wherein n is a positive integer, the coefficients of terms equally distant from the beginning and end of the expansion are equal.

3. By Art. 1, the coefficient of the (k+1)th term is „C Therefore, by Ch. XXX., § 3, Art. 4, we have:

The greatest binomial coefficient, when n is even, is „С„; and when n is odd, is „C2-19="Cn+1°

4. In (1 + x)" = 1 + „C1x + „C2x2 + ··· + „С„x”, let x=1.

Then

That is, the sum of the binomial coefficients is 2".

That is, the total number of combinations of n things, taken one at a time, two at a time, and so on, to n at a time, is 2′′ – 1.

That is, in the binomial expansion, the sum of the coefficients of the odd terms is equal to the sum of the coefficients of the even terms.

§ 2. BINOMIAL THEOREM FOR ANY RATIONAL EXPONENT. 1. From Ch. XXVIII., Art. 4, we have

(1 + x)" = 1 + +()x+()*+

(1)

when n is a positive integer. In this case the expansion ends with the (n+1)th term, since the coefficients of the (n+2)th and all succeeding terms contain n − n, or 0, as a factor.

But

if n be not a positive integer, the expression on the right of (1) will continue without end, since no factor of the form n-k+1 can reduce to 0. Therefore this series will have no meaning unless it be convergent.

2. In Ch. XXXIII., Art. 27, it was proved that the series

is convergent when x lies between 1 and +1. It remains to be proved, therefore, that in this case the above series represents (1 + x)”, when n is a fraction or negative.

3. Since the reasoning will turn upon the value of n, we shall call the expression

a function of n, and abbreviate it by f(n), for all rational values of n. To understand the following reasoning, the student should notice that for all positive integral values of n, (1+x)" = f(n), as, (1 + x)3 = ƒ(3) ; and that it remains to prove that (1 + x)" = f(n), when n is a fraction or negative, as, for example, that (1 + x)3 = ƒ(}).

...

f(x)=1+ (H) x + (H) x2 + + (x^1)22 + --

for real values of x between 1 and +1.

= 1

We will assume that the product f(m) × f(n) is a convergent series, when the two series are convergent. The proof of this principle is beyond the scope of this book. We then have