28 



1+2 + 4 + ... = -!, 



while by others they are regarded as meaningless results, and 

 have thus been altogether rejected from analysis. 



It is impossible to avoid the occurrence of these series : 

 they present themselves at a very early stage of algebra, in 

 the form of geometrical progressions and binomial develop- 

 ments ; and thenceforward are continually met with by the 

 analyst up to the remotest applications of the integral calculus. 

 The existing vagueness and indecision, as to the proper mode 

 of interpreting such series, is thus a matter of some concern, 

 as calculated to retard the progress of science, to diminish our 

 confidence in some of the truths of analysis, and to give cur- 

 rency to results involving error and contradiction. 



In the present communication it will be my endeavour to 

 ascertain the causes of the perplexities and discrepancies above 

 adverted to, and to discover the legitimate interpretation of 

 diverging infinite series ; from which it will, I think, follow 

 that certain expressions received into analysis as the sums of 

 several of these, are erroneous. The fact that Poisson, Cauchy, 

 Abel, and indeed most of the modern continental writers, re- 

 ject diverging infinite series, and pronounce them to have no 

 sums, does not render such an endeavour the less necessary ; 

 inasmuch as the analytical operations, in virtue of which finite 

 values have been attributed to extensive classes of these series 

 by Euler and subsequent investigators, remain, I believe, un- 

 impugned. Widely diff"erent methods appear to concur in 

 furnishing the same numerical results for such series ; as, for 

 instance, the method of definite integrals, and that deduced 

 from the difi"erential theorem, both so frequently applied by 

 Euler to effiect the summations of series of this kind ; and the 

 numerical results obtained by him have often, apparently, been 

 verified by later computers ; some of whom have employed 

 methods quite distinct from those of Euler ; as, for instance, 



