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

CHAPTER IX.
APPLICATION OF THE DIFFERENTIAL CALCULUS TO THE DEVELOPEMENT
OF FUNCTIONS; WITH SOME PRELIMINARY REMARKS RESPECTING SERIES.
WE now proceed to shew some of the principal uses
which may be made of the differential calculus in pure
mathematics, and in the first place its application to the
developement of functions in series,
125. It will be necessary to make a few preliminary Prelimremarks -on the nature of an ordinary series of the form mark respecting
o, + A- + Am2... +A +- &&c....; series.
and to settle precisely what we mean by the equation
f(x) = oA + A1x + A2,2...+ An,"+ &c....(1)
which expresses the developement of a function f(x) in a
series of powers of x.
126. In the first place we may remark that the sign= always
=here, as well as elsewhere, always signifies actual equality; denot
for mathematicians always consider themselves at liberty, when equality.
they have two expressions of any kind connected by the sign
=, to use them indifferently for each other, and to substitute one for the other in any- calculation; which certainly
they have no right to do, if the sign = does not always
signify actual equality. We shall therefore always use = as
the sign of actual equality.
127. Secondly, by the second member of the equation (1) We must
not in gene
we do not in general mean a series of terms infinite in number, sal suppos,
but a series of terms carried on according to a certain law a series to
consist of
to any number we please, with a term at the end of a dif- an infinite
number of