194 



Algebra. 



V 



19. Binomial Thkorkm for any Exponent. 



Let us apply Taylor's formula to the development of (x-\-h)" 

 according to positive increasing powers of //, where ;/ is either 

 positive or negative, integral or fractional. 



Here, then, f(x^h) = (x^-h)\ 



Therefore f(x)-=x". 



Finding the successive derivatives oi x" we obtain 

 D,/f.i-; = ;2.r"-', Y)lf(x) = 7i(n—\)x"-\ 



J}lf(x)=^n(n—i)(n — 2)x"-\ J}^.f(x)=-n(n—i)(n — 2)(n — 2,)x"-' 



etc. 

 Therefore by substituting in Taylor's formtfla we get 

 , , , ii(n — I J , 71(71 — "i-jfii — 2 J 



^,^(n-^n-^(n-^)^,,_^^ ... (I) 



li 



Thus we arrive at the Binomial formula, where, however, the 

 exponent is not restricted, as in Chapter X^, to being a positive 

 whole number. 



From this we see that the Binomial formula in its greatest gen- 

 eralit}^ is a special case of Ta3dor's formula. 



The series (i) will be finite if 71 is a positive whole number, but 

 not otherwise. 



When the series (i) is infinite it should be examined to see 

 whether it is convergent or divergent, for values 7nay be given to 

 -v, //, ;/, which will render the equation (i) untrue. 



For example, let x=i, /; = — 3, and ;^= — 2 ; then the left-hand 

 member of (i) becomes (1 — 3)"^ which equals (— 2)"% which 



equals — — , which equals \, a definite quantity. 



But the right-hand member of (1) becomes 1+6 + 27 + 108 + . . 

 a sum of positive whole numbers each greater than the one be- 

 fore it, and evidently the sum does not approach \. 



20. Examples. 



J. Develope (i—x)~'^ by Taylor's formula. 

 2. Develope (i—x)~'^ by the Binomial formula. 

 Compare (i) and (^) with XII, Art. 8. 



