of the Sums of Neutral Series. 137 



is less than 1, it is still equal to -— . I am not sorry to have the 



m 



opportunity afforded me of expressing more fully my views 

 upon this point. 



It is somewhat difficult to conceive how, by considering the 

 series 1 — 1 + 1 — &c. as the limit of another series, or by 

 considering it in any other point of view, we can make its 

 value different from what it is. If it be the limit of the series 

 1— x + x* — &c, where x is less than 1, and if moreover the 



limit of this last is — , it follows incontrovertibly that 1 — 1 



m 



+ 1— &c. must in all cases = — . The mistake here arises 



2 



from calling 1 — 1 + 1 — &c. the limit of 1 — x + x 2 — &c, where 

 x is less than 1. It is no such thing. It is indeed the value 

 which that series assumes when the limiting value is given to 

 the variable ; but it does not thence follow, nor is it the fact, 

 that the one series is the limit of the other. We might ex- 

 pect the case to be otherwise, but it is not. Prof. Young 

 himself admits, that without exception 



X^ + ^+^^^^^+i^^; . .(«.) 



and this holding always will hold in the limit when a: = 1, 

 which gives us, when n is infinite, 



] _ i x i _ & c . = — + — = I or indifferently. 

 2 2 



From the same original equation we likewise deduce this 

 other, that when x is not greater than 1, 



1— a? + a- 2 — &c. = — — , 

 1 +x 



except in the limit : whence it follows that the series 1 — a? + a; 2 



— &c, where x is not greater than 1, approaches — ■ as its 



limit. Now this is not more incontrovertible than that 1 — 1 

 + 1 — &c. is equal to 1 or 0, from which it is evident that the 

 series 1 — a-+a? 2 — &c, where x is less than 1, does not approxi- 

 mate to the series 1 — 1 + 1— &c. as its limit; for the limit of 

 a quantity or ratio is that quantity or ratio to which it conti- 

 nually approximates, and from which, although it never actu- 

 ally reaches it, its difference can be made less than any as- 

 signable quantity. It is perfectly true then that the limit of 

 the series 1— x + x 2 — &c, where x is less than 1, or of the 

 series 1 — (1 — x) + (1 — ar) 2 — (1 — x) 3 + &c, where * is greater 

 Phil. Mag. S. 3. Vol. 28. No. 185. Feb. 1816. L 



