Mr. DE morgan, ON DIVERGENT SERIES. 185 



on this equation, l'^ — 2' + ... =0, Abel, another rejector, remarks (Works, li. 266), " Peut-on rien 

 imagine!- de plus horrible?" 



Poisson's mode of allowing ^ = 1 - 1 + ... is clearly equivalent to an adoption of the maxim 

 that whatever is true up to the limit is true at the limit. When relations of pure magnitude 

 are in question, there is no doubt of the truth of this principle. But the words iip to must not 

 be understood inclusively, since then the principle would merely assert that what is true at the 

 limit and elsewhere, is true at the limit. With this caution, it is impossible to prove that a relation 

 of magnitude is true at the limit, if at the limit we have no longer calculable magnitude. We may 

 not say that what is calculable up to the limit is calculable at the limit, nor that what is complicated 

 up to the limit is complicated at the limit, &c.: but only that relations which are quantitatively true 

 up to the limits are so at the limits, if the limits be quantities. Assume 1 — 1 + ... to be quantity, 

 determinate quantity, a«d that quantity may possibly be shown to be J and no other: but it 

 may not be assumed that 1 - 1 + ... is a quantity, because 1 -^ + ^- ... is a quantity, up to 

 its limit; or at least if such assumption may be made, no reason has been given for confining 

 it to any one class of limiting forms. 



Again, it is clear enough from the manner in which Fourier, Poisson, Cauchy, &c. use the 

 limiting form 1 — 1 + ..., that they intend it to signify ^ in an absolute manner. The whole 

 fabric of periodic series and integrals, which all have had so much share in erecting, would 

 fall instantly if it were shown to be possible that 1 — 1 + ... might be one quantity as a limiting 

 form of Afj-A-i+... and another as a limiting form of B„-B^ + .... Fourier's celebrated 

 expression of a function by means of a definite integral, that of Poisson by means of a series 

 of periodic integrals, &c., are all stated as absolute truths, and used as such, though they are proved 

 only as limiting forms of one particular class of convergent series. A person who is much versed in 

 the writings of the above-mentioned analysts must feel to his finger's ends that one well-established 

 instance in which I — \ + ... means other than ^ would throw doubt upon all they have written. 

 Now we have Poisson's assurance that these series, though indeterminate, have each a unique value, 

 which can be employed in analysis when the series are considered as the limits of convergent series. 

 Here the word 'indeterminate' is loosely used, in the sense of not determinable by actual summation : 

 a unique value, which can be employed (and therefore of course first found) is not indeterminate in 

 any correct sense. But who is to assure us of this uniqueness of value.'' How could Poisson 

 undertake to make the assertion.? By an induction — an extensive one I grant — but still an induction. 

 From (1 + j)-' = 1 - X + ... to* 



:/ 



J ^. g-2i) VIogx ■ 



it is always observed that where the series-side of an attainable developement gives I — 1 4- ... the 

 finite side gives ^. But this induction may be overturned : and if the stability of form which really 

 has hitherto characterized series of finite divergency should be found not to belong equally to those 

 of infinite divergency, it .should teach us rather to suspect the former than to content ourselves with 

 merely empirical rejection. There are two ways of considering a series : absolutely, as a given 

 algebraical expression, and relatively, as the development of a given function, from which it 

 actually was pi'oduced. I do not defend the former mode of considering either convergent or non- 

 convergent series ; and I fully believe that analysts have been led into error, as to both classes, by 

 incautiously reasoning on series of which the invelopments were unknown. I do not dispute that 

 the arithmetical value of a specific case of a series may, when that particular case is convergent, be 

 calculated : but, speaking of general scries, it seems to me that it is dangerous to reason upon them 



* This imunce in very good lor the purpose, nincc one «idc or the other must have all the difficulties of divergency : either the 

 integml or the Meries in divergent. 



