184 mh. de morgan, on divergent series. 



a paru croire qu'une telle serie peut etre employee dans les calculs analytiques a la place de 

 la fouctioti ; et quoiqiie cette erreiir soit loin d'etre generate parmi les geometres, il n'est 

 cependant pas inutile de la signaler, car les resultats auxquels on parvient par Tintermediaire 

 des series divergentes, sont toujours incertains et le phis souvent inexncts." 



Pages 408, 409. "On peut voir dans les Memoires de Petersbourg (^Novi Commentarii, torn. 

 XVII et xviii) la discussion qui s'est elevee autrefois entre Euler et D. Bernouilli au sujet des 

 series de sinus ou de cosinus prolongees a I'infini. Les details dans lesquels nous venons d'entrer, 

 ne semblent devoir laisser aucune obscurite sur ce point d'analyse : nous admettrons avec Euler 

 que les sommes de ces series considerees en elle-memes, n'ont pas des valeurs determinees; mais 

 nous ajouterons que chacune d'elles a une valeur unique et qu'on peut employer dans Fanalyse, 

 lorsqu'on les regarde comme les limites des series convergentes, c'est-a-dire, quand on suppose 

 implicitement leurs termes successifs multiplies par les puissances d'une fraction infiniment peu 

 differente de I'unito." 



I hardly know which of the passages in my Italics ought to excite most surprise. Divergent 

 series, at the time Poisson wrote, had been nearly universally adopted for more than a century, 

 and it was only here and there that a difficulty occurred in using them. As to the second 

 passage, we may clear Poisson of absolute mistatement by remembering that he had both head 

 and hands full of a subject which had tasked his great powers to their utmost, namely, the 

 substitution of definite integrals for series in questions of mathematical physics. As far as in- 

 tegration is concerned, I admit, and even think I shall presently show, that he was fully justified 

 in what he said : in the meantime I attend to his argument in favour of finitely diverging series. 



Let us take the series 1 — 1 + 1 — 1+..., a remarkable specific case of both algebraical and 

 trigonometrical series. I collect from what I have quoted, and from numerous other parts of 

 his writings, that Poisson is content to equate^ to 1 — 1 +... , considering the latter as a mere 

 form indicative of 1 - g + g'— ... , where ^ is a fraction infinitely near to unity, but less. He 

 will consent to use the limiting form of convergency, to walk on the line which separates con- 

 vergency from divergency, but not to cross that line, even by an infinitely small quantity. 



In using the language of infinitely small quantities, I do not intend to direct any part of 

 my argument against the ideas connected with the phraseology, because both Poisson's statements 

 and my comment on them might easily be translated into the language of the theory of limits. 

 Let us then take 1 — 1 + 1 - ... as indicating 1 -g+g'-—... where 1 - §■ is infinitely small and 

 positive. How can 1 — g + g'- ... be called convergent? Because the terms diminish without 

 limit, and g", if n be injinitely great, becomes infinitely small. The departure from finite 

 divergence, and commencement of real convergence, is infinitely distant. Now all that is 

 wanted to make 1 + 2 +4+... equal to — I is the presence of the infinitely great negative re- 

 mainder, which might be considered as not destroyed, but only removed, when the second side 

 of (1 - 2)"' = 1 +3+ 2"+... + 2" + 2" + ' (1 - 2)"' is made an infinite series by n = co. If sup- 

 positions which only take effect at an infinite distance from the beginning of the series are 

 allowed to be made with regard to series of finite divergence, why may not the same be conceded 

 in the case of infinite divergence.'' Both 1 — 1 +... and 1 +2 + ... are equally irreducible to 

 their finite equivalents by the arithmetical computer ; both are equally creatures of algebra : if 

 a reason can be shown for the distinction between them, those who adhere to infinitely diver- 

 gent series have a right to ask for it ; but if, as I suspect, that reason be experience, I am 

 prepared to contend that, when integration is not employed, there has not been produced one 

 single instance in which divergency, properly treated, has led to error. 



That experience is the guide may be safely inferred in all cases of rejection, when those 

 who reject do it to different extents. Poisson would admit 1^ — 2* + 3^ — 4^ + ... =0, since there 

 is no question that, g being less than unity, the mere arithmetical computer might establish, 

 to any number of decimal places, the identity of 1^ - Z^g + 3^ - ... and (1 - g) {\ + g)''. But 



