Cre 
* 
33 
employed is always necessary to such completeness. This 
has already been seen in the development of or(l—z)"’, 
l—z 
which is a particular case of the binomial development: be- 
a 
t— 
sary to the complete algebraical equivalence of the two mem- 
bers of the equation. And it is plain, from the nature of 
common division, that a like supplementary addition must be 
made to the infinite series furnished by the development of 
1 
l 
sides the series, the supplementary expression r is neces- 
- ——— or(l—2)-". In the extraction of roots, too, as in 
(1—2)" 
(1—z)', (l—2)}, &e., it is equally plain that, however far 
the extraction be extended, we approach no nearer to the ac- 
tual exhaustion or annihilation of the algebraic remainder; 
and therefore we are not authorized to dismiss this remainder 
and to account it zero, when general algebraic accuracy is to 
be exhibited ; although, as in geometrical series, we may do 
this in those particular numerical cases in which the remainder, 
if retained, would vanish. It thus appears that, calling the 
remainder after n terms, whether n be finite or infinite, f(z), 
the ordinary binomial series, to n terms, will be the complete 
development, not of (l— 2)"; but of (l—a— F(a)"; and 
therefore that, if this series be equated to ( 1—2)" merely, it 
will require a supplemental correction to produce strict alge- 
braical equivalence; which correction must be such as to 
vanish for those numerical values of x, which cause f(x) to 
vanish. 
These values are all those which render the series diver- 
gent: for, as well known, we can, in every such case, approach 
by the series alone as near to the numerical value of the un- 
developed expression as we please. It is thus only when the 
series ceases to be convergent, that the correction adverted to 
has any arithmetical existence, adjusting the equality of the 
