Infinite Series* 15 f 
2. If the feries tfv 0 H-^+ s + ar 0 + 2 ” + &c. converges, then will 
the feries a' x x + b f x x ~ n + c / ^ x " 2 ” + &c. diverge, unlcfs in cafes of 
the flowed: convergency, where x = -±zV y/z±=i 9 and all the 
roots of the given equation are of the formula ±v — 1 ; 
when v = ±v \/ — 1, the fucceffive terms of the infinite feries 
deduced from algebraical quantities by the preceding method 
will ultimately, that is, at an infinite diftance come more near 
to a ratio of equality with each other than any aflignable 
difference* 
3. If the fluxion be + + . x rri ) m xx 9 — J x, their 
will A = mrn + 6 , and x' = 9 - rn. 
Many more propofitions concerning infinite feries and their 
convergency are given in the Medit. Analyt. 
4* Let the fluxion be a + bx n Xx n x 9 of which find the fluent 
in a feries afeending according to the dimenfions of and it 
will be a m x x b +* ^ 
m 
4 - m . 
m — - 1 
b~\-i l 
m— 2 
X -#" + *».. 
m- 
2 {b + 2 n -f 1 ) 
X - . .v :s 
3 (A + 3 «+ i) 
,3 ,« + I 
x — xi n + &c.) = — 4-1 
mb r 
\ — 1 
/I Vt » ■ 
/ft . 
x 
mn-{-b-\~ 1 — 
4 See. —a-i-hx"' 
mn-\-b -f J — * 
h + ~i i.h+i+n'* + £+1 . A+i-M • ^ + r + 2# 
b* v v ^^_ 2/!+I (w+ 1 + * (?» + + 1) . + J ) x | 
h+ 1 . ~b + 1 + w . /? + 1 + 2« . h + 1 + 3« 
X 2 
£ 3 
L^+ 3 «+* 4- &c.) = (bx n 4- a) m + l x ( T 
e* / \ / \ (.z 
£ 4 - 
na 
mn + b-\- l . 0w4*^ + I —nb 
[mn-^b + i)b 
X X h *'~ zn + 
X * — i 
b-\- 1 — n xb+ 1 — 2 » X /2 
-p 6 4* iX mn + h 4 i — nXmn + X — 2»^ s 
xy j + l " M -&c. (L). The 
fi rft 
