494 
note on Lagrange’s demonstration oe taylor’s theorem. [536 
to as many terms as we please we obtain a result differing as little as we please 
from f(x + h). Under these circumstances then we may assert the truth of Taylor’s 
theorem.” 
I share Abel’s horror of divergent series ( 1 ), and I maintain the validity of Lagrange’s 
demonstration. When by an algebraic process we expand a function in a series, for 
instance the function , by division 
JL CO 
1 — x) 1 (1 -f- X + X 2 -p &c. 
1 — X 
X 
X 2 &c. 
in the series 1 + x + x 2 + &c., and write accordingly 
—-— = 1 + X + x 2 + &c. 
1 — X 
all that is (or ought to be) meant is that the algebraical operations continued as far 
as we please will give the series of terms 1, x, x 2 ,... or say the series of coefficients 
1, 1, 1,... And of course with this meaning of the equation, the objection “non 
constat that the series is convergent” would be wholly irrelevant, we do not say that 
it is, we do not care whether it is so or not. In further illustration, remark that 
we frequently use such an equation merely as the means of expressing the law of a 
series of numbers a 0 , a u a 2 ,..., say a n = coeff. x n in f(x), where the function is assumed 
to be by a definite process expansible in the form a 0 4- a x x + a^c 2 + &c. in question. 
Any objection that the series is not convergent would be simply irrelevant. Now any 
rational or irrational algebraic function f(x + h) can by ordinary algebraical processes 
be expanded in the form f(x) + terms in h, h 2 &c.... And if in regard to a function 
fix) we make the single assumption that f(x + h) is expansible in a form containing 
powers of h and reducing itself to f(x) when h is put =0, then Lagrange’s demon 
stration shows that the powers of h are h, h 2 , h 3 , &c.... and that the expansion in 
fact is 
f(x + h) =f{x) + hf (x) + ~f" (x) + &c. ; 
viz. f(x + h) acquires the same value f(x + h + k) whether we change therein x into 
x + k or h into h + k ; and the expression on the right-hand side is the only series 
in h possessed of the same property. It is to be remarked that the equation contains 
in itself the definition of the operation of derivation, viz. the equation being true, 
f (x) can only denote the coefficient of h in the expansion of fix + h) ; and what 
1 Peut-on imaginer rien de plus horrible que de débiter 
0 = 1« - 2 n + 3” - 4« + etc., 
n étant un nombre entier positif?—Œuvres, t. n., p. 266; [Nouv. Éd., 1881, t. n., p. 257].