589 



Suppose Ihe series (25) to converge for & "^ 6^ and to diverge 

 for 0<:^(l^, and, coiiseqiientlv, the series (26) to converge for 

 R (.<•) > /' -|- 0^, and to diverge for R (.<■) < /' + ^>i, then the integral 

 (1) will, at any case, not admit an expansion into a series of 

 factorials for any value of A' (' ) <C '' + ^A • ^^ "ow shall prove 

 that for any positive ^S, taken as small as we please, 



SO that A < ;.' -j- (7, ; by this the i-eipiired proof will have been 

 established. 



For the sake of brevity we write 



t: ^ 6^ -{- ó = u. 



Consider the derivative of negative oriler — « of (i {t), which 

 according to the definition of Riemann ^), is given by 



t 







then If' (0 'S ^ function regular at ^ = with the same circle of 

 convergence (0,1) as v (0 has; its expansion into a power-series is 



t|HO=TT. " — 7T (27) 



From this formula it may be derived that if^ (/) remains unite for 

 i=:i, in virtue of the initial hypothesis. 

 Conversely we have 



First, let 



Then we may choose (f so small that also « <C J and write 



rf (t) = D- ^ D[t-M^) V) = D- ^at- ^x\it) f t-iY\t)\ =z 



1 / ! • (28) 



= ..Vr- -. \{t-n) '■'\uMic)n-^^ + if''(«)u«| dn 

 . ƒ (I -n) ) ] 



(I 



Now If' {u) is, in the range ^ ?/, < 1 , finite and thus less than 

 a certain number (j. Hence 



1) Or for negative values of A' 4- 6^, Urn (1 - <)> +9i4-"-|-''> "0 (t) = 0, if n is such 

 that A' + Ö1 + « > 0. 



') See among others Borel, LeQons sur les séries a termes positifs, p. 75. 



3) Properly speaking it should be D.Z)«-i, but this operation, in the present 

 case, is equal to !>« i . D, since the subject of the operation is zero for t = 0. 



