and on Fourier's Theorem. 485 



But if we adopt a different local arrangement of the two 

 series, we have 



S=l-l + l-l + &c., 

 S =l-l + l-l+&c.; 

 whence we get 2 S =2— 2 + 2 — 2+ &c. in inf. 

 = 2 or indifferently. 



Now there is no reason whatever for preferring the first of 

 these methods to the second, and in fact the two results are 

 equally admissible; whence it is plain that 2 S is actually 

 more* indeterminate than S, having the three values 2.1.0, 

 and does not by any means enable us to get rid of the ambiguity 

 of value of the latter quantity. 



I have entered into the consideration of the above method of 

 investigating the value of the series 1 — 1 + 1 — &c., on account 

 of its being somewhat difficult for the student to discover the 

 fallacy of the common mode of conducting it. But another 

 and more satisfactory method may be given as follows : — 



We find by actual division, that 



1 ^ — r'i'^+l 



l_A^ + a;2-^-3 + &c. +{-'x)n = -i L-^Z- . 



^ 1 + ^ 1 -\-x 



If ar be less than 1, the second member of the right side of 



the above equation ^— — vanishes when n becomes infi- 



' 1 +.r 



nite, in which case we have 



1— A' + .r^— ^+ &c. in inf. = . 



\ + X 



If j; be greater than unity, when ti becomes infinite the same 

 quantity becomes indefinitely great, but there is nothing in 

 the circumstances of the problem to determine its sign (since 

 it is impossible to say that an infinite number is either odd or 

 even). The true conclusion therefore is, that in this case 

 either sign will serve, or that 



i—x-\-x'^—ar^-'r &c. in inf. = + go indifferently. 

 Lastly, if A'=l, we have 



* The rationale of this result is obvious. We may take 1 or for the 

 value of each series at pleasure, and there is nothing to restrict us to take 

 the same value in the two cases, much less are we bound because we take 

 the first vahie 1 in the first case to take the second in the other, as is ac- 

 tually done in the common method. Hence to obtain all the values of the 

 sum of the two series, we must combine each possible value of the one with 

 each possible value of the other, which gives us the three values 2.1.0. 



