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

.92

FAILURE OF TAYLOR S SERIES.

and it is easy to see that neither d (z) nor any of its differential coefficients become infinite when = 0: therefore we
have
f~m) = (0) + ) + q2(0) F2 +)3(o) + &c.
ri rz 2
By actual differentiation, and putting z = 0, we find
tP (0) = sin 2, q'(0) = 0, 02 (0) = 2cos 2, 0'(0) = Gcos 2...&C.
and therefore,
sin IM + 1 + (x-1)1} =sin2+cos2.(x-1)+cos2.(w-1)1+&c.
Same ex- This method however is not generally the simplest in
ample
done more practice; other substitutions for iv often bring out the result
simply. more readily. Thus assume
iF -1I+ ("I - = Z
and then fa(x) = sin (2 + z) = (z), suppose;
and it is evident that neither cP (z) nor any of its differential
coefficients become infinite when F = 1; i. e. when z =0; we
have therefore
x x2~~~~
P (z.) = P(O)$ ~)(0) l + c'(o(0)+ + &C.
Now since ~(z) = sin (2 + 4 we have
4P(O)=sin2, qp(O)=cos2, c2(O)=-sin2,
(0) =-cos 2... &c.;
and therefore, restoring for z its value, we have
f(iF) = sin2 + cos2. {(I - I) -1)+ (i. -..
if we arrange this in powers of (x - 1), expanding each term
by the binomial theorem, we find the developement required.
ff(e) being 143. These methods clearly suppose that f(a) is not
infinite indicates the infinite.