Al Fete 
where H is a certain positive number greater than G. For R («) > 
R(e) the remainder has therefore zero as a limit as 7 increases 
indefinitely. Moreover the majorant-value (20) shews that on the 
half-line going from #-=c in the direction of the positive part of 
the real axis, the binomial series converges uniformly; for R (#—c) = 
x—c on this line. PiIncnERLE has observed (l.c) that a similar state- 
ment, which is analogous to a known theorem of ABEI. on power 
series, holds for the integral (1), and that it follows from the 
equation (5), which has been found by means of integration by 
parts. In the same manner the just mentioned proposition may be 
proved generally by means of summation by parts, both for series 
of integral factorials (the binomial series treated of in this note) and 
the series of factorials in the more restrictive sense of the word, 
which proceed®according to inverse factorials. For the latter series | 
have shown this in a communication on those series *). The expansion 
of the integral (1) in such a series is, however, as appears from 
investigations of NIELSEN *) and PiNcHerLeE®), only possible under 
restricting conditions for f(é), viz. if it is an analytic function, whose 
circle of convergence for the point £1 passes through ¢ = 0, and 
whose order on this circle is different from —+ oo. 
1) Proceedings XXII, NO. 1. 
*) Handbuch, p. 244. 
3) Sulla sviluppabilità di una funzione in serie di fattorali, Rendic. d. R. 
Acc. d. Lincei 1903 (2e Semestre). 
Proceedings Royal Acad. Amsterdam. Vol. XXII. 
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