436 



NiKLSEN. There is not any simple indication that it would always 

 be satisfied. Only in the case of absolute convergence of the series 

 of integrals for u = l, i.e. when H[x) ^ I' -\- 1 , this is realized, 

 because the series (20) then converges at the same time uniformly 

 in the closed interval 0<n<\, and therefore represents, according 

 to a well-ltnown proposition, a continuous function of u in that 

 interval '). 



') The reasoning of Pincherle, who, as already remarked, treats of absolute 

 convergence only, seems to be incorrect to us. it is based upon his writing for 

 the separate integrals of the series (20), if we replace ii by 1 — c, 



1— £ 



(1— f"+')rj/ 



ƒ 



.r(.r+ 1) . . . (ic 4-w) 

 

 which is wrong. If this could be done, the limit of the series (20) would indeed 

 be equal to the series of the individual limits of its terms, if the latter converged, 

 even if this convergence took place only conditionuUy. It seems therefore not 

 superfluous that, in the last sentence of the present paragraph, we have 

 called attention to the strong way in which the idea of integration term by term 

 may be used to prove the possibility of developing the integral (1) into an aöso/i<^e/y 

 converging series of factorials. 



