348 ALGEBRA.
is referred to Chapter xlv. It is there shown that when x
is less than unity, the formula
n(:)i-l) ., nhi-l)(n-2) , (1 + xy = 1 + nx + — ^^T^-^'-v- + ^ -^J^3 -^ of^ + -
is true for any value of n.
When n is negative or fractional the number of terms in the expansion is unlimited, but in any particular case we may write down as many terms as we please, or we may find the coefficient of any assigned term.
Ex. 1. Expand (1 + x)^-3 to four terms.
1-2 1.2- 3 = l-Sx + 6x^- 10x3 + ... 3
Ex. 2. Expand (4 + 3x)3 2 to four terms.
oTi , 3 3x , 3 9x2 1 27x3 , "1
422. In finding the general term we must now use the formula
7i(7i - l)(n - 2) '•' (n - r --l) ■ r ^'^
written in full; for the symbol nCr cannot be employed when n is fractional or negative.
Ex. 1. Find the general term in the expansion of (1 + x)^1 2.
The (r+1)th term
1 2 (1 2-1)(1 2-2)(1 2 - r + 1) x^r, r
= 1(-1)(-3)(-5) ...(-2r + 3) x^r, 2r r
