Summation of Series. 387 
ef equal magnitude, as +eand « ai/-1, &c. andx=a; 
in which cafe fometimes both fericfes may become the fame 
converging feries, &c. 
“When wx, in the preceding cafes, is equal to the leaft or 
greateft root, the feries will fometimes converge, and fome- 
times not, as is fhewn in the above-mentioned chapter. Whe- 
ther the fum of a feries, whofe general term is given, can be 
found or not, will in many cafes appear from the law of the 
multinomial and other more general feriefes. 
“2. There are feriefes which always converge, whatever may 
be the value of x; as, for example, the feries I oe 
I 
me tale 
ee may be the ae of «3 but it may be obferved, that 
thefe feriefes never arife from the expanfion of algebraical 
functions of x, or the before-mentioned fluents; but, ina few 
cafes, they may from fluxional equations. ‘There are alfo 
feficics| Which “mever converge as 1 ft 3x31 0 2. on 
+1.2.3.4%'+&c. to which the preceding eee may be 
applied. 
3: In the year 1754 fome papers, which contained the firft 
edition of my Meditationes Algebraic, were fent to the Royal 
Society, in which was contained the following rule, viz. let S 
be a given function of the quantity x, which expand into a 
feries (2 + bx" + cx" 4 &c.) proceeding according to the dimen- 
fions of «; in the quantity S, for x" write ax”, Bx”, yx", &c. 
where «, @, y, &c. are roots of the equation x7—1=0; and 
let the refulting quantities be A, B, C, D, &c. then will 
A+B-+C+D-+&c. 
eee be equal to the fum of the firt, 27+-1, 32+1, 

Pa * 4 8ec.ort + ar ;+3 4+ &c. &c. always converge, 
&e. terms 7% infinitum. ‘This method, in the preface to the 
Eee 2 Jatt 
