S64 Prof. J. Jl. Young o« ^*e Evaluation of 



nexion with the convargent csLse^ of tnat series; out the fijj^me 



^ 



extreme ciise may be regarded as lirarting the diTiergenl cases, 

 and therefore — its connexion exclusively with these cages 

 b(.'in;>- symbolized — as identical with ",' '^^ '*■"'"'' "' '"" 



JSd tHfttii*easoning as before, and recollecting that \\-\ ) 



== ^, and that e'^' is infinite, we infer that the limiting case of 

 tJie series of divergent cases is, like every one of the case§ 

 which precedes it, infinite. 



But the series 1—1 + 1— 1+ &c. is frequently met with in 

 analysis uncontrolled by any law binding it in connexion with 

 a continuous series of either convergent or divergent cases, 

 and where, in consequence, it would be unwarrantable to re- 

 place it by either of the forms employed above, since in these 

 the fact of such connexion is, as an essential condition, im»- 

 pressed upon each ; and it is from overlooking this circum- 

 ptance that the series in^ciint irfr Ai'dii&c. is so often errone- 

 ously assumed to be the representative! of —. As an illustra- 

 tion, suppose the summation of the infinite series , , 



2 3 4 5 , o ^ 



-lui I)bi? {I "as. 5 5.77.9 9.11 ' ^38^3 j^ni^isv 



'were required : the true value would be obtained' by'siibtiract- 

 ing the lower of the following series from the upper, and then 

 dividing the remainder by -fif^^-^^^^i; A iioUiicl ahl aioliid baji, 

 2 3 il-^riw f« J'JGTJsdB jiodii js t'^4"8I 

 mil JR Siwvi o8lft-wjfj-»* ^cf^{— ei«w^ ►f' &Cir;oqa'i badaildnq 

 i« badisiidoq ai bmi ,«;;*^' ..Mnul^ UkoA daril Ijs^oH 



_^ ^ _l_ _1 _ &^'3'i^t ^^^^ ni jrfjgaoi 



<i(j (li d^^noffjlB Jfidi ,3J;;.5 '» ' v ''^<t jifik viau jib^ ^qfirf-taq gi Jl 



+ — — -, seeing that the lower series must always project 



one term to the right beyond the upper. Now if we replace 



the series [2.] by — , twioeithe sum of the proposed series 



' iif] - 2^ ^"'%— ' jiliw aofxonno;) ^ii lo ghBm 

 will be ambiguously — ■ — ^ + ^ ; 'which 'is absurd, 



It is plain that the sum of the series [2.] is to a certain ex- 

 tent indeterminate, being indifferently 1 or 0; and on the 



