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.

91

Thus it appears that when any differential coefficient
fP(a) becomes infinite, it indicates the appearance of a fractional power in the developement off(x) in powers of x - a.
142. We may generally apply Taylor's series, in the How the
following manner, to determine the developement off(x), when deelopeto
fractional powers appear in it in consequence of some of its be obtained
when any
differential coefficients becoming infinite when x = a. of the differential
coefficients
Put x = a + z', and f(x) will become a function of z, beomenin(p(z) suppose: then, if possible, so determine r that neither finite.
(p(z) nor any of its differential coefficients shall become infinite when z = o, and this being the case we shall have, by
Taylor's Theorem,
(Z) = gp(o) ' + 02(o) 1... n () r + R,
n being as large as we please.
Now here put for z its value ( - a)y, and for (P(z)
its value f(?), and we have
(x - a);
I
= +! (0) (a a) ( a)
+ p'(O) 172 p5(o ) + R,
which is the developement required containing fractional
powers.
There are also other ways of substituting for i, so as
to obtain the developement of f (x) by Taylor's series.
Ex. Let f(i) = sin {I + 1 + (a - 1)}; Example.
here it is easy to see that f2(o) = co; therefore we shall not
be able to develope f(v) in this case in integral powers of
- 1. To obtain the developement by means of Taylor's
Theorem, put = 1 + 2, and then
f(v) = sin (2 + z' + 03) = (j),