433 



denoting tlie rest of the series of" factorials in question, lias zero as 

 a limit for 71 = co, if R {.v) ^ '/.' -\- I , and thus that for these values 

 of .t' development of the integral (i) in such a series is possible. 

 That this series for these values of .v converges absohitely, may, as 

 we had already occasion to reinark, as well immediately be derived 

 from 3a'. 



4. We make the following remarks: i**'. The proposition proved 

 just now may be compared with a result due to Cesar^ ^), according 

 to which it follows from the condition 3a" that the inecpiality (17), 

 for Iünt = i, is already valid from and after the value n = 0. 



2"^^ In our result, and in that of Cksaro, is included that the 

 number X, introduced by Niet,sen, is for the natural majorant of 

 ff{t) exactly equal to X' -\- J "), whereas for the function itself it is 

 in any case not greater; thus we have 



X<X i- I, and I^X' + 1 . . , . . . (18) 



if A is the number in question for fp(t). This result, derived by 

 Nielsen from his fJieoreni, can tlierefore, if our opinion with regard 

 to the inexactness of Nielsen's proof is right, no more be regarded 

 as deduced by him. 



3"^. We may also easily prove that inversally every series of 

 factorials such as (6), if it converges at all, is equal to an integral 

 of the form (1), where <f (t) satisfies the conditions under 1" and 3a". 

 In fact, if such a series converges for a certain value a -j- ii-i of .<■, 

 the limit of the terms must be zero for that value. Now we have 



nf 1 



lim — — 



H= 00 .'?'(« -f 1) ••• (^ + n) n« 



from which it follows that the coefficients a,, of the series of 



factorials satisfy the condition 



lim a,i -^ n^ , 



where X' is a certain real number which is at most ecjual to <i, 

 but may be equal to — cc. If now we form with the coefficients 

 an a function ff{t) as in (2), this function has the properties expressed 

 in the conditions 1° and 3a", and, as we proved in the foiegoing 

 paragraph, the given series of factorials is equal to the integral 

 (1) for R{x) > ;.' + 1 ; at least, if A' + 1 > 0, for if not, we should say 



1) Ace. d. Scienze fisiche e malemfltiche di Napoli, 1893. See also Borel» 

 Legons sur les séries a termes positifs. 



') At least when a» has a "croissance reguliere" ; otherwise it may be smaller 

 than A'+ I, but no smaller than A'. 



