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

TAYLOIR'S SERIES. 8

83

Lemma XXI. fails when any of the quantities f (a),. f '(a),
f '(a),.f (a) become infinite, this developemient fails under
the same circumstances.
This developement is known by the name of Taylor's Taylor's
1) ~~~~and MaeSeries; when a = 0 it is called Maclaurin's Series, in which1auns
case it becomes series.
it J~~~~~2.f () = (0)+f I(E2 Enf2 En
We may write the developement, (1) thus:
f (a +h) =f (a)+f(a)-~ f 2(a) +. +fn(a)- + &c.
remembering what "1+ &c." means.
The following are very important examples of this. de- Examples.
velopement.:135. Let f (v) = x'1; and therefore Deve'lope(lnt)of
then putting 1 for a in the developement (1), we find
(I+ht + -nh + h a,
1
+ E~~n
which is the binomial theorem for any value of the index,.
Let f (x) =logv, and... f Qv =x F(n - 1) ( );then- Developeputting I for a in the developement (1), we find letogf1+)
1 h2 h3 hn
Let f (X) O X, and f~ f" Gv) -=knox, where k =log c, thenr Developenient of
putting a = 0 in the developeme'nt (1), we find
kh k'h2 knhn
Ei E2 En &C