RATIO. 



fideralion of ratios whicli have limits is difficult, we (hall 

 begin with examples of other quantities. 



I ft. Let there be formed a feries, whofe firft term is i ; 

 fecond, i ; third, ^ ; fourth, a, and fo on ; every term 

 being half the preceding one, -viz. 



1 I I I I 



1, — . — . -jT' -7. — . Sec. 



2 4 8 i6 32 



and let the fum of an indefinit number of terms in this 

 feries be confidered as continuali, increafed by the accellion 

 of a new term ; thus, the fum of t. le two firfl is i^, of three 

 terms is if, of four terms is I J, :kc. I fay then, that the 

 varying fum of the terras of this ii ties continually approxi- 

 mates to the fixed number 2, as its limit. For the difference 

 between i, 2, 3, &c. terms and the number 2, will be the 



numbers l, — , — » -a , —,, &c. fucceflively, in in- 

 2 4 8 16 ' 



finilum. Here it is- evident, that the terms in this laft feriesj 

 which exprefs the fuccelfive diilereiices between the increaf- 

 ing fum of the former feries and the number 2, ^r//, con- 

 tinually decreafe ; and fee ond/y, no term in this feries of dif- 

 ferences can become either notiiing or negative ; and thirdly, 

 we may continue this leries of fuccefTive differences, till we 

 arrive at a term which fhall be a lefs part of the fixed num- 

 ber 2, than any fractional part of it that can be afligned ; 

 or fo that this difference (hall be lefs, when compared with 

 the number 2, than any ratio affigned. The number 2, 

 therefore, having the conditions laid down in the definition, 

 is to be called the limit of the fum of the terms of the 



Tor the terms of this feries; will be /, 



a~ 



infinite feries i, 



— , &c. 

 16 



1 I 



2 4 



And the fame is to be underftood of any other infinite 

 feries ; "uiz. if a number can be found having the above con- 

 ditions, that feries is faid to have a limit ; and the finding 

 of this limit, is what is to be underflood when mathema- 

 ticians fpeak of finding the fum of fuch an infinite feries. 



No number lefs than 2, for inftance i|, can be taken 

 for the limit ; for, in this cafe, it will not anfwer the 

 fecond condition of the definition. In the above example, 

 the fum of four terms of the feries is equal to i^, and 

 the fum of five terms exceeds it ; therefore, the difference 

 between this fum and the number 1 J propofed as a limit is, 

 in the former cafe zero, and in the latter negative. Nei- 

 ther can any number greater than 2, as for example 3, be 

 taken for the limit, becaufe here the laft condition will be 

 wanting ; for if the fum of any afligned number of terms 

 be lefs than 2, that fum mull always want more than 

 unity of the number 3, and confequently cannot approach 

 nearer to 3, than any afligned quantity, as i. 



In like manner .the fum of the feries 1, — , — ~, 



3 9 27 

 &c. continued in ttifimtmn, will be i^ ; the feries of fuccef. 



five differences being ~, -^-, -g, —, Sec. in injnitum. 



Now, in order to make the preceding example general, 

 kt a:i exprefs the common ratio of any feries of numbers 

 in continual proportion, whofe firft term is unity, I fay 

 then, if a be greater than i, fuch a feries will have for its 



limit the quantity —— ^ ; or, in other words, the fum of aU 



the terms of fuch an infinite feries will be " 



T, &c. 

 Whence the fum of i term is 

 of 2 terms 



of 3 terms 

 of 4 terms 



a -)- A 



a 



a'-\-ai + b' 



a' 



a^ a^ b -^ ab'- -^ b' 

 a' 

 Let each of thefe fums be fubtraded from the limit 



, and we have the fucceflive differences ( \ 



"-'' \a-bJ' 



/ '' ±\ {J_ l'^\ / b b^\ 



\a-l • a J' \a-b ' >/' V^ITa " IT/' ^^- °' 

 if H be any afTigned number of terms, the differeHce be- 

 tween the fum of that number of terras and the limit 



abb"'' 

 __, will be —J . -;p-^. Whence we may obferve, ift, 



that as the number of terms whofe fum is required increafes, 



this difference continually decreafes, becaufe -- , being a 



fraftioB lefs than unity, its powers continually decreafe. 

 2dly. This diff"erence can never become notiiing or negative '■ 

 the powers of a fradion, though they decreafe, being aK 

 ways real and affirmative. 3dly. This difference may be. 



come lefs in refpeft of -^, than by any alHgned ratio. 



For 



as a" : i" ; or as ^ to I ; or as 



b a — b a 



\b) " to I : and fince -^ is greater than i, and n inde- 

 terminate, the former term of the ratio may become greater 

 than any affigned quantity, and therefore the ratio itklf lefs 

 than any ratio affigned. 



The quantity y^, having, therefore, the required condi- 

 tions, is the limit of the above feries, or is equal to the fum 

 of all the terms continued in infinitum. 



What has been proved above may be fhewn more concifely 

 by dividing a by a - b, in the manner of divifion irt algebra, 

 for the quotients will be the very feries propofed in the ex- 

 ample : tor inftance, 



a-b) a (i + --f-— -i-—-j-&c. 

 a — b '^ <i a 



b^ 

 a 



a' 



And 



