1822.] Mr. Marrat on Neutral Series. 417 



Article VI. »S^ 



Some Observations on Neutral Sei'ies. By W. Marrat, AMi. 

 Member of the Philosophical Society, New York, and private 

 Teacher of Mathematics, Liverpool. 



(To the Editor of the Annals of Fhilosophy.), 



SIR, 

 The following neutral series 1—1+1 — 1+ &c. to infinity 

 has exercised the talents of some of the greatest mathematicians 

 since the time of Leibnitz, who was among the first that noticed 

 it. Dividing 1 by 1 + «> we obtain the following series, viz. 

 1 — « + a^ — «^ + «* — &c. ; and by taking the value of a 

 equal to 1, it produces the series of units above given, because 

 every power of unity is 1 . Leibnitz makes the above series, 

 when « = 1, equal. to ■^, and \\\ that one pairtieular case it is true. 

 Euler notices the same series in his algebra. " This series," he 

 observes, " appears rather contradictory ; for if we stop at — 1^ 

 the series gives nothing, and if we stop at + l,it gives 1. But 

 this is precisely what solves the difficulty; for since we must go on 

 to infinity without stopping eitlier at + 1 or — 1, it is evident 

 that the sum can neither be nor 1, but that the results must lie 

 between the two, and, therefore, be equal to ^.'^ In the caser 

 before us, where the series is deduced from the expression; 



r , it becomes, when a = ] , — — - =1 — 1 +1 — 1 + &g. 



.... and the series is evidently equal to ^ . The result, as 

 Euler observes, certainly lies between and 1, but it is not uni- 

 versally equal to ^* for ■ ~:4-=l — 1 + 1 — 1+ 8cc 



gives also the same series as that above for -i; and again 

 - , V . =4-=l — 1 + 1 — 1+&C which is still the 



1 + 1 + 1 4^ 1 ^ . . , , 



same ; and the series will be the same whatever may be the 

 number of ones in the denominator of the fraction from which it 

 is produced. The arithmetical mean then between 1 and 

 only gives the value of the series in one particular case out of an 



infinite niimber. Affain, we have - — ; — r = 4= 1—1 + 1 — 1 + 



• o/ i+l-f-1 -^ 



&c or /t^t^. = f = 1 - 1 + 1-1 +&C. .... and 



1+1+1+1 ^ 



the same series will always be produced if the number of o?/es in 



the denominator exceed tbe number in the numerator ; that is^ 



the series will be produced, if nine units be divided by ten units, 



or 99 units by 100, 6r 999 by 1000 ; whence it is obvious that 



the series is equal to any fraction less than unity, which is its 



New Series, vol. iii. 2 e 



