OF AN INFINITE ANGLE. 259 



\ — ,1: + ,1.2 _ ... by writine: 1 for .r the sum is A. Now let us turn the fraction '- '-^ 



•^ ■^ 1 +X + X-+ ....v''-' 



into a series by the ordinary process of division ; the result is, (6 > «) 



1 + 'V + ... a terms , , .^ 



= 1 -*'"+. r*- .i;'> + » + .r2''- 



I + o) + ... b terms 



This series differs from 1 — ,v + a:^— ... only in being more general, for it includes it as a particular 

 case (viz., when a = 1, and b = 2). If then it be lawful to write 1 for a; in 1 — ,t' + x-— ... it 



is equally lawful to do so in the more general case : which being done we have — = 1 — 1 + 1-.. . ad 



btfinUvm. Here then is "a well-established instance in wliich 1 — 1 + 1 — ... means other than 1;" 

 shall we say witli Professor De Morgan, one such instance throws "doubt on all that Poisson and 

 Fourier have written ?" 



6. It will hardly be considered necessary to defend a system which requires us to receive as a 



legitimate consequent that all proper fractions are algebraical equivalents. I apprehend therefore 



the last article but one will be sufficient to shew that in numerical forms of series the ability of an 



e.xpression to furnish by legitimate expansion a proposed series is no presumption that the two 



are algebraically equivalent. Hsie then is fair ground for suspecting the existence of some grievous 



violation of just reasoning in depriving an infinite series of its remainder, i. e. in supposing that by 



pushing an expansion in iiifinitum the anomalous terms may be disregarded. In converting the 



1 + 1 + 1 + ... o, terms . . ,,.,,, 



expression . into a series we observe that for all values of a and b (a <b) the 



'^ 1 + 1 + 1 + ... 6 terms ^ ' 



series of quotients are the same, and the various cases are distinguishable only by their remainders. 



The distinctive properties then of these proper fractions by the process of development are not 



thrown into the quotients, but are preserved in the remainders. How then shall we reject the 



remainders in any equation which professes to exhibit the equivalence of its members .' 



But there is yet another proof, which I shall now offer, that neither 1 nor even any proper 



fraction whatever can be the proper equivalent of the series 1 - 1 + 1 — 



7. In perusing what has been written upon this series, we cannot but perceive that some authors, 



.1 



setting out with = 1 - ,r + x" as an equation admitted on all hands to be true when w is 



I + w ' 



less than 1, have argued that, being true when x is less than 1, however small \ — x may be, it must 



needs be allowed in the limit. If the premises are true, I do not see how we can refuse to allow the 



conclusion. But it is obvious the premises assume that the series is convergent towards 1 - 1 + 1 - ... 



when 1 - ,1; is indefinitely small ; is this true ? If it is, I admit that i is the equivalent of the series 



1 - 1 + 1 - ... in as good a sense as is the equivalent of the converging series 1 — x + or- ... 



Mr. De Morgan questions this; but I see no objection in it which would not, if admitted here, 

 overturn the whole fabric of the Differential Calculus. But we have to answer the question asked 

 above, is it true that the series I — x + x- -*•' + ... is convergent towards 1 - 1 + 1 - ... as its 

 limiting form when 1 - ,c is indefinitely small .? 



8. Let y be any finite quantity, and assume 1 — .t? = ± - : then when n approaches infinity, 



I - X will be indefinitely small ; but tiicn limit of .r" = limit of [ 1 =f - J = e^", the upper or lower 

 fiign being used according as x approaches 1 from inferior or superior values. Here then is a proof 



L L2 



