Infinite Series , 
*5jr 
when ; s a whole negative number ; the latter when 
is a whole affirmative number. 
In like manner affume for a descending feries (A* 8 -J- x 
{aV+ I “' 1 + ( GV+ I ~ 28 -f ^V+ I “3« 4. &c.)j or (bx n - f x 
(A#~" w+I * 4 - + Ca;“ , ”+ 3» ^ & c .) ; the former will termi- 
nate when is a whole affirmative; the latter when 
n 
rnn 4 - h +_ i j s a whole negative number. 
« 
It appears, therefore, that a feries will terminate equally 
by an afcending or defcending feries ; and the end of the one 
afcending feries is the beginning of its correfpondent defcend- 
ing one. All thefe feriefes, which do not terminate, proceed 
on in infinitum ; one term in the former feries becomes infinite, 
when lo + i +zn = o ; and in the latter (L) when mn 4 - A 4 * i 
zn~o 9 z being a whole affirmative number. 
io. It has been obferved in the Medit. Analyt. that if fome 
quantities contained in the given irrational ones are much lefs 
or greater than the reft, it may be preferable in the former 
cafe to reduce them into feriefes not proceeding according to the 
dimenfions of x 9 but according to the dimenfions of thofe 
quantities; and in the latter Cafe according to the reciprocal 
dimenfions of them ; and particularly fo if the fluent or inte- 
gral of the terms of the refulting feriefes can be found in finite 
terms, or by tables already calculated. 
From fimilar principles to thofe before given may be found 
when the refulting feries will converge, and when not. 
This method will in many problems be ufeful, when the 
value of a near approximate is known. 
Vol* LXXXI. Y 
Of 
