430 



that, if A<^;/-|-1, there is a certain domain on the left of 

 R{j;) =z 'a' -\- 1, where the series (6) converges conditionally. If 



A' <;.<;'+ 1 



this convergence takes place in a strip of the plane determined by 



A < R(,v) <X + I 

 and if 



in one determined by 



r < i2(.r) < ;.' 4- 1. 

 This is remarkable, becanse Pincherle ^), who has also published 

 some notes on series of factorials, does not mention the case of 

 conditional convergence. And in order to state at once our point of 

 view, we declare not to understand the reasoning of Nielsen from 

 which the conditional convergence within the domain mentioned 

 would follow; it is even our opinion, as we are going to explain 

 direclly, that from his investigations onlj- the validity of his statement 

 follows for the values of.?; satisfying both the conditions /^(,r) '^ ^ -\- i 

 and R{.v) "^ X -\- 1 . The example given by Nielsen of the develop- 

 ment of an integral such as (1) in a conditionally converging series 

 is exact, and it is not difTicult, as it will be seen, to add others to 

 it. But in reflecting on the subject we have finished to doubt of the 

 general validity of the special theorem of Nielsen, in any case, for 

 the strong proof of it investigations of a farther reaching extent 

 would, in our opinion, be necessary. 



2. Tiie mode in which Nielsen arrives at the development (6) 

 consists in integrating by parts the integral (1) : this gives iramediatelj' 



X 



a. 



(m — l).^a„_i 



R„= r(;r) f "^"^^^^^^; '^[^^ —dt . .... (11) 



X x{x-\-\) x{x-\-\)...{x-\-n — 1) 







where 



1 



(1— «)^+"-i 



'{x^n) 







From the condition 2° Nielsen derives that the first of the ine- 

 qualities of (5) will be valid for an interval ö<:^t<^l, \f Rix)"^?.. 

 Here the quantity <f may be conceived arbitrarily small ; it must of 

 course be understood that the choice of the integral number N is 

 influenced by it, and that N will increase indefinitely with 1/d. 



') Rendic. d. R. Ace. d. Lincei (1903, 2e Sem.). 



