434 



that a certain remniruler of the given series is equal to an integral 

 of the form (8) in the footnote of p. 429. 



4. If the coeflicients a„ in the power-series for (i{t) are real, we 

 can assign one detinite case in which the series converges c'f/i(^//;ViV/ia//v 

 for R{x) ^ //, viz. if all derivatives of an order higher than some 

 definite number have the property of conserving the same sign 

 throughout the interval 0<^<[1, and this in such a wajMhat always 

 two immediately succeeding derivatives are of opposite signs. This 

 result can be deduced from the equality 



i2,,-7?„+i =————-— (19) 



.«(.r + l)...(;c + n) 



where R„ has the meaning given by (IJ). First let x be equal to 

 a real number rr. Then R,i and — Rn-\-i 'lave the same sign throughout 

 the interval of t, so that either of them is smallei', in absolute 

 value, than the series-term in the right-hand member of (19). If 

 therefore the latter has zero for its limit, then also R,„ and this is 



the case for « ^ /', since lim a,, -^ w' . 



If X is complex ^ a -\- i^ and a ^ >/, we have 



1 



r/("(<)(l — 0^+"-i 





dt 



■+l)...(.r-t-n-l) 

 



1 



^1 «(f(f l)...(« + n-l) /• </(")(«) (l—0-+"-i 



<^ I I at. 



^ « + z/J)...(«-f i/i-f n— 1) IJ «(.« + l)...(« + n— 1) 







The latter integral is equal to R^ for x = n and therefore, as 

 we showed, zero for n = QO\ the factor by which this integral has 

 to be multiplied evidently has a modulus smaller than unity ; thus 

 /?„ approaches to zero as n increases indefinitely. 



As an illustration we take the example of Nielsen 



\i-tY-^ 



ai.)=j^^ 



dt, 







where 



ifit) = — , ;i ^ — 00 , ;.' = 0. 

 1+f 



This function 'f{t) satisfies the S|)ecial condition mentioned in the 

 present remark, and on account of this circumstance we may deduce 

 the possibility of developing the function rz(.f) into a series of factorials 

 for R(x)^)/, this series converging but C(>??c//^/o>2<7//y if /i*(.*') <^ /'-)- J . 

 Other examples to illustrate his theorem, especially such that do Jiot 



