NATURE 



97 



THURSDAY, JUNE 4, 1903. 



INFINITE SERIES. 

 Th^orie t^Umentaite des Sdries. Par Maurice Godefroy : 



avec une Preface de L. Sauvage. Pp. viii + 268. 



(Paris: Gauthier-Villars, 1903.) Price 8 francs. 

 TNFINITE series present themselves in mathematics 

 J- in diflferent contexts, serve different purposes, and 

 admit of different interpretations. The simplest case is 

 when, from a numerical sequence («i, u.^^ «3, . . .), we 

 derive the series 



«l + «2 + ^3 + • • • 



which we may denote by "Zu. It is assumed that there 

 is a rule for calculating «„ when n is assigned ; if we 

 write j„ for «i + «2 + • • • + «»» there exists a sequence 

 (jj, J.2, J3, . . .) and we may, in fact, regard 1u as being, 

 in a manner, a symbolical expression of this sequence. 

 When we say that 2« is convergent and its sum is j, 

 what is really meant is that the sequence (x„) converges 

 to the limit .v. 



To Cauchy and Abel is mainly due a strict theory of 

 such arithmetical series. They showed that, whether its 

 terms are real or complex numbers, a series of this sort 

 may be divergent, indeterminate, or convergent ; and 

 that series which are absolutely convergent may be com- 

 bined by processes which we may call addition, subtrac- 

 tion, multiplication, and division. There is one part of 

 this theory which, even yet, is not always made so clear 

 as it might be. Suppose that we have two sequences 

 («;i), {"^'n) of such a character that every element u^ of the 

 one occurs as an element v,^ in the other, and conversely ; 

 that this is a (i, i) correspondence, that is to say, that 

 each element of one sequence is associated with one, and 

 only one, of the other ; and, finally, that when/ is finite, 

 q is also finite, and conversely. In this case we may call 

 (t/„) a permutation of (7/,,). When 2«„ is absolutely con- 

 vergent, so is 22/„, and the sums of these two series are 

 the same ; it is this property, really, that makes abso- 

 lutely convergent series so easy to work with. Properly 

 speaking, a series is distinct from its permutations ; but 

 in the case of an absolutely converging series this dis- 

 tinction may be ignored. It is a remarkable fact that a 

 series and one of its permutations may both converge and 

 have different sums. It is rather unfortunate that the 

 phrase "changing the order of the terms in a series" is 

 still used ; it is certainly best to regard a series as 

 defined, not merely by its terms, but by the order in 

 which they are written. 



After discussing this arithmetical theory, M. Godefroy 

 proceeds to the next simplest case, when the terms of the 

 series are functions of a variable x which is supposed to 

 assume numerical values. Here the distinction between 

 uniform and non-uniform convergence appears, a dis- 

 tinction first emphasised by Stokes and Seidel. In the 

 sequence (j„) derived from a convergent series of this 

 kind, the index n for which j„ first differs from the sum 

 of the series by less than an assigned quantity h is, in 

 general, a function of x as well as of -* ; so that for par- 

 ticular values of x and their immediate neighbourhood 

 n may be enormously large even for values of h which, 

 though small, are not infinitesimal ; accordingly the 

 NO. 1753, VOL 68] 



series is no longer available for practical calculation. 

 At such places the convergence ceases to be uniform ; 

 the convergence is uniform wherever it is possible to- 

 assign, in terms of h but not of .r, a value of n for 

 which I J„ - -y I < h. 



Of course, the most important series of this class are 

 power-series, and in his third chapter M. Godefroy deals 

 with them at some length. On pp. 67-69 he gives 

 Dirichlet's proof of Abel's fundamental theorem that 

 when a power-series is convergent its value at the 

 boundary of its circle of convergence is the lim.it of its 

 value as x approaches the boundary. To learn to appre- 

 ciate the necessity for proving this theorem is a good 

 exercise for the mathematical student ; it looks sa 

 obvious and is yet so far from being a truism. 



The remaining three chapters are on the exponential 

 function, the circular functions, and the gamma-function 

 respectively. The noteworthy features are that sin Xy 

 cos X are defined by power-series, that the transcendence 

 of e is demonstrated, and that the properties of the 

 gamma-function are deduced, after the manner of Gauss» 

 from the product n («,;r). M. Godefroy points out that 

 Weierstrass's formula 





was explicitly given in 1848 by F. W. Newman {Camb^ 

 and Dubl. Math. Journ.^ vol. iii. p. 59). 



The final chapter is the one which presents most 

 novelty in the shape of actual results ; thus, besides the 

 series of Stirling, we have various interesting formulae 

 due to Prym, Hermite and others. But M. Godefroy's 

 style and method will attract the reader's attention 

 throughout ; he combines simplicity with rigour, and is 

 neither dry nor diffuse. His work is one which may be 

 cordially recommended, especially to mathematical 

 students ; not the least of its merits is its excellent 

 bibliography, which is just what a treatise of this sort 

 should contain. 



M. Godefroy does not explicitly introduce the complex 

 variable, but it is easy to complete the chapter on power- 

 series so as to make its results apply when x is complex. 

 Thus we have, on the whole, a discussion, with illustra- 

 tions, of numerical series, and of power-series which 

 define functions of a variable within a circle of con- 

 vergence. 



Incidentally, we have examples of two other kinds of 

 series. Stirling's formula is the classical example of a 

 series which does not define a function, but which, while 

 ultimately divergent, serves to calculate the numerical 

 value of a function very exactly for any sufficiently large 

 value oi X. Such asymptotic series have been recently 

 studied by Poincard, Borel and others, and their properties- 

 are no longer a mystery. 



Again, Lambert's series 



X X^ X" 



l.X + l-X^+ ■• -^ i-xn+'-- 



is an example of a series which serves for enumeration. 

 If each term is expanded in powers of x, and the result 

 collected, we get 2\//'(«).i", where •^(«) is the number of 

 ways of solving n = 8B' with integral values of 8, 8', the 

 order of 8, 8' being taken into account except when they 

 are equal. Thus yj^(n) = 2 when n is prime, but not 



F 



