8. Find the conditions that a^2x^4 + bx^3 + cx^2 + dx +f^2 may be a perfect square.
9. Prove the identity a^2(x-b)(x-c) (a-b)(a-c)} + {b^2(x-c)(x-a) (b-c)(b-a)} + {e(x-a)(x-b) (c-a)(c-b)}
FUNCTIONS OF INFINITE DIMENSIONS.
485. If the infinite series a_0 + a,x + a.x*+ a,x°+--- is equal to zero for every finite value of x for which the series is convergent, then each coefficient must be equal to zero identically.
Let the series be denoted by S, and let S_1 stand for the expression a_1 + a_2x+a_3x^2+; then S=a+aS_1, and therefore, by hypothesis, a)_0+ xS_1 = 0 for all finite values of x. But since S is convergent, S_1 cannot exceed some finite limit; therefore by taking x small enough, xS_1 may be made as small as we please. In this case the limit of S is a_0; but S is always zero, therefore a_0 must be equal to zero identically.
Removing the term a_0, we have xS_1 = 0 for all finite values of x; that is, a_1 + a_2x+a_3x^2+ vanishes for all finite values of x.
Similarly, we may prove in succession that each of the Coefficients a_1, a_2, a_3, is equal to zero identically.
486. If two infinite series are convergent and equal to one another for every finite value of the variable, the coefficients of like powers of the variable in the two series are equal.
Suppose that the two series are denoted by
a_0 + a_1x + a_2x^2+a_3x^3+ and A_0 + A_1x + A_2x^2+A_3x^3+ ;
then the expression
