VOL. LXXIV.] PHILOSOPHICAL TRANSACTIONS. 5S7 



tween a and — «; but if a? be greater than a or — a, the above-mentioned 

 series will diverge. Let -k be the greatest root of the above-mentioned resulting 

 equations ; then a series descending according to the reciprocal dimensions of x 

 will converge, if x be greater than + n ; but, if less, not. When impossible 

 roots a + b</ — 1 are contained in the equations, an ascending series will con- 

 verge, if x be less than the least root + a, and + (« — b) and + (a -f- b) ; or 

 more generally, if a: be less than the least root + a, and x"+ T at an infinite dis- 

 tance n, be infinitely less than 



2a«- 2 .n.—~ a-»i s + Z.n*-— . —1 . —2 a"~H* - &c. 

 2 2 3 4 



(a 2 + b 1 )" ' 



a descending series will always converge, when x is greater than the greatest root 

 of the resulting equations ; and x"-', w/hen n is infinite, is infinitely greater 

 than (a + b)" and (a — b)" ; or more generally than 



la- - in . "-^- a"-*b' + 2n. ^"— . ?-=-? . '— a" - *b* - &c. 

 2 '234 



This follows from Caput 3 of the Meditationes Algebraic*. 



Cor. It hence appears, that if the ascending series converges, the descending 

 will diverge ; and, vice versa, if the descending converges, the ascending will 

 diverge, unless all the roots of the above-mentioned resulting equations may be 

 deemed of equal magnitude, as + « and — <*., a.^ — 1, &c. and x = a.; in 

 which case sometimes both serieses may become the same converging series, &c. 

 — When x, in the preceding cases, is equal to the least or greatest root, the 

 series will sometimes converge, and sometimes not, as is shown in the above- 

 mentioned chapter. Whether the sum of a series, whose general term is given, 

 can be found or not, will in many cases appear, from the law of the multinomial 

 and other more general serieses. 



2. There are serieses which always converge, whatever be the value of x ; as, 



for example, the series 1 + ~ x + ^J^—^ + &c - or J + £ + % + % + 

 &c. &c. always converge, whatever may be the value of x ; but it may be ob- 

 served, that these serieses never arise from the expansion of algebraical functions 

 of x, or the before-mentioned fluents ; but, in a few cases, they may from 

 fluxional equations. There are also serieses which never converge as 1 + 1 

 .' 1x -f- 1 . a . 3x 2 +1.2.3. 4x 3 -\- &c. to which the preceding remark may 

 be applied. 



3. In the year 1757 some papers, which contained the first edition of my 

 Meditationes Algebraicee, were sent to the r. s., in which was contained the fol- 

 lowing rule, viz. let s be a given function of the quantity x, which expand into 

 a series (a + bx m + cx lm + &c.) proceeding according to the dimensions of x ; 

 in the quantity s, for x m write <x.x m , (Ix™, yx"", &c. where a., (3, y, &c. are roots 

 of the equation x" —1=0; and let the resulting quantities be a, b, c, d, &c. 



4f2 



