vanishes for more than n values of x; and therefore, by the preceding article, that is,
Hence the two expressions are identical, and therefore are equal for every value of the variable. Thus
If two rational integral functions are identically equal, we may equate the coefficients of the like powers of the variable.
Cor. This proposition still holds if one of the functions is of lower dimensions than the other. For instance, if
we have only to suppose that in the above investigation
q_0 =0, q_1 =0, and then we obtain
484. The theorem established in the preceding article
for functions of finite dimensions is usually referred to as
the Principle of Undetermined Coefficients. ‘The application
of this principle is illustrated in the following examples.
Ex. 1. Find the sum of the series 1-2+2-3+3-4+ + n(n+1).
Assume that
1-2+2-3+3-4+ + n(n+1)=A+ Bn+ Cn^2+ Dn^3+ En^4+ , (1)
where A, B, C, D, E, are quantities independent of n, whose values have to be determined. Change n into n +1; then
(2)
By subtracting (1) from (2),
This equation being true for all integral values of 2, the coefficients of the respective powers of n on each side must be equal; thus E and all succeeding coefficients must be equal to zero, and
3D=1; 3D+2C=3; D4+C+B=2;
whence D=1 3, C=1, B=2 3.
