\o\t> the Sums of Neutral Series^ ^oiH 363 



■•►'flr^ — j;^ 4- Scc^iifw to yidi'l .ibaonhnoa) la«<mK 



the accuracy of which expression, under all circumstances as 

 to the values of .r and «, this latter being a whole number, is 

 universally admitted. It is also allowed, when n is infinite, 

 that all the convergent cases of the series, short of the extreme 



case X =: }, are comprised in the single fraction . Now 



although I hold it to be an axiom, that if the expression suf- 

 fice for all cases, short of the extreme case, it must of necessity 

 suffice for that too, yet it may be well to show the entire con- 

 sistency of this truth with the general expression [1.] for n = 

 infinite. 



In order to this, let 1 j- be put for a; ; then it is plain 



that we shall commit no error if for w + 1 we write k. co ' ; so 

 that the sura of the infinite series 



;!:(!_ n + u_n^ _ (,_>)% (,_ ^ r- 



o; \s:\ ; -*^. v ^ f"^ ^ ^ //u? 1o no»fh9JJP*i4irp 

 w|U b^ <;orreGtly expressed by- - > i ^ f ' —' v^ arfj -l 

 ,,;,; ijrio /(Til?, 1 \ ^>t.c»<.irMDob ^m 



;.r.' s = ^ _iZ±—llL^ . ' yn-u,. 



,,.,fT i . tm. vi4* ll-'-^); il^liil + C 1 t) 91 



whatever be the value of it. f ^ 



Take now the extreme case,"^ia:'<36 , which in feet is tharw^ 

 a?=l, and we have ,^1^ 



' ,^_, {-o-i)r-°°; ^....:i 



But it is well known that ( 1 ) = — , and that ( — j >'. 



= 0, consequently i) vjiip'lq-i^;^ aUom.j^nlfB.Mr set 



^ ~ ■ 7 Tx 7 p\ T" 2 ' .«>?wiini!ft>i 

 14.(1 ) i + (i ) 



The extreme case of the general series l — a: + x^— a-^+'Scc., 

 which we have here been considering, the case namely of 

 *= 1) has been considered exclusively in reference to its con- 



