An elementary treatise on the differential calculus, in which the method of limits is exclusively made use of, by the Rev. M. O'Brien.

88

TAYLOR'S SERIES.

By (131) we may assume that
f(x) = Ao + A (x - a) + A, (x - a)2... + A ( - a)n + R....(),
A0, A...An being any arbitrary constants, and R that quantity,
whatever it be, which must be added to the second member of
the equation to make it equal to the first. If we can so
determine AO, Al,,A...A, that the limiting value of
(a? - a)"
when x approaches a shall be zero, then the developement (1)
will be a perfect developement. Now if this limiting value
= 0, it follows from Lemma XX. Cor. 3, that
dR d2R d3R d 7R
dox ' d32 dx3
must be each zero when x = a. Hence, differentiating (1)
put in the form
f(x) - Jo- A, (- a) - A,) (a - a).... - A (-a) = R
n times successively, and then putting x = a, we obtain the
following equations:
f(a) - Ao =
f'(a)- rl. A = o
f2(a)- rF2. A = 0.........),

fn(a)- Fn.A = o
which equations give us the values of A,, A,, A... A,,
necessary to make (1) a perfect developement.
And conversely, if we give the values (2) to Ao, A,,
A2.. Ab, we shall evidently have
d dR dR dnR
R = o, d 0' = ~ -'d =, when x= a;
r drf<2 d x"

and therefore the limiting value of (- --. when x approaches
(X - a)"
a, will (by Lemma XX. Cor. 2.) be zero; i.e. the developement will be perfect.