33 



employed is always necessary to such completeness. This 

 has already been seen in the development of t— — or {\—x)-\ 

 which is a particular case of the binomial development: be- 

 sides the series, the supplementary expression — — is neces- 

 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 



or (1 — a;)-". In the extraction of roots, too, as in 



{l-xf ^ ' 



(I— a;)*, (1 — a;)*, &c., 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, /(a;), 

 the ordinary binomial series, to n terms, will be the complete 



development, not of (1 — re)'", but of (1 — a; — /(a;))'" ; and 



therefore that, if this series be equated to (I— a;)'" 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 a;, which cause fix) 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 



