

402 Dr. Warine on the 
is the term inthe given feries, whofe dienes from the ff 
is z. Inthe fame manner may be deduced the fum of a feries,. 4 
whofe general term is fx soranzsx 7’ -s-ns, or by re- 
peated operations 7’ x ex" -+-fe+g, where ¢’ is a term of the | 
given equation, whofe diftance from the firft term 1s 2. And 
_ in general, from the fum of a given feries, whofe fluxion can. 
be found, and whofe general term 1s #’, can be deduced by con- ; 
tinued multiplication, and finding the fluxion, the fum of a 
feries or quantity, of which the general term is A”, where A) 
is any function of the following kind a’2” 4-6’2"—" 4 clam 4. &c. 
in which z denotes the diftance from the firft term of the feries,, 
and mawhole number. It isto be obferved, that if the given. 
feries converges in a ratio, which 1s at leaft equal to. the ratio: 
of the convergency of fome geometrical feries, the refulting 
equation will always converge. But if in a lefs ratio, then 
it will fometimes converge, fometimes not, according to the 
ratio which the fucceflive terms of the refulting feries have 
to each other at an infinite diftance. 
Cnilarys 2 Bea sai Boni o. Bee ees 
8, YToPPEL AX EQ er Howe OPIS 
PES. PRT. PRS 25 PHSTP 2 Zr. aol ee i r 
PS PET . TPS eR ee p-I » if pr bein wiole 
affirmative number; but this latter quantity has the formula; 
above-mentioned az” + 4z"-* +cz"-3 + &c. ; and confequently,. 
if the fum of the feries a+ dx + cx + dv3' + &c. =p be known,. 
by this method can be deduced the fum of the feries) 

a2 by pie 1 t oxtt B Pett? dst 4 &e. | 4 
: le ae ix Oe | 
Ex.1.Sincea +x" =a (142x243 xB 7 ait } a 
m—2n 

4 
a-2x° +&c.); multiply the fucceflive terms bf this feries 
inte 
x: 

