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

FAILURE OF TAYLOR S SERIES.

89

Hence it appears (putting x - a = h) that
f(a) +f'(a) +f2(a)......" f (a) + R
is a perfect developement, and the only perfect developement,
of f(a + h).
We must except however the case where any of the
quantities f(a)(a), '(a), f2 (a)...fn(a) are infinite, for then
we cannot be sure that the equations (2) are satisfied by
giving A,, A,, A... &c. the above values; for instance, if
f3 (a) = -, we cannot assert that f3(a) - F73 3 = 0, if we give
A, the value -, for then the quantity f3(a)-F3. A, assumes
1 1
the form - —, which we cannot assert is zero.
0 0
140. If f (t), and all its differential coefficients below The failure
a certain one, the pth suppose, be finite, but the pth (and all of Taylor's
series.
above it by Lemma XXIII.) infinite; then the developement
(1) holds as a perfect developement for all values of n less
than p, but for all other values of n it fails.
141. When a failure of this kind takes place, it is A differengenerally possible to obtain a perfect developement containing tiaelobeff
fractional powers of h after the pth term. coming infinite when
x= a, indix - a cates the
For let F(x) =f(x) -f(a) -f'(a) appearance
r1 offractional
powers in
(v -a)2 (w a\'P-\ the devel-f I(a)(..a"p - (), see (123) opement
r2 F(p-1)
then it is clear that
F (a) = o, '(a) Fo, r(a))= o.... - (a) = 0,
but FP(a) =fP(a) = co; since we suppose that f(a), f'(a),
f2(a)....f-(a) are finite quantities, but fP(a), fP+'(a)..
infinite.