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

78

NATURE OF A SERIES.

terms, but ferent kind from the rest, which is commonly called a reof a finite
number mainder: the former terms are represented by
with a remainder. A, +- Ax + A2,... + Anns,
n being any integer, and the remainder is represented by the
" + &c.," which is written after the term Anx". The remainder we shall often denote by writing + R instead of " + &c."
Thus when we say that
1
= + t + X2...+ Vn +-t &c.......(2)
we do not in general mean to assert that 1 + x + x2... to
1
an infinite number of terms - for if so, suppose = 2
1 —,O'
and then we have 1 + 2 + + +16,.. to an infinite number of
terms =- 1 which is manifestly absurd. But we simply mean
to assert that
1 ----- = 1 X4< +.......... (3)
1 - So
where n is a number as large as we please, and R a certain
quantity which must be always added to the series to make
the second member of the equation equal to the first. In fact
the " + &c." in equation (2), and the + R in equation (3)
mean the same thing.,n+1
In the present instance the value of R is -, for it is
easy to see that
1,'1n+1
1 +, +..x+
1 -a? 1 --
and thus when x = 2 we have
- 1 = 1 + 2 + 8 16... 2n - 2S1
in which equation there is no such absurdity as there is when
the number of terms is considered infinite.

We may
Under cer

128. Thirdly, if it should so happen that R is a quantity