286 NEW SERIES for the 
manifeft, that each term of the feries in our fecond formula 
mutt be lefs than = of the term before it, but greater than a 
third proportional to the two terms immediately preceding it ; 
and thefe are the limits to the rates of convergency of our fe- 
cond feries. 
19. WE may alfo affign limits to the fum of all terms, after 
any propofed term: for putting it under this form 
An 
> 
Le | 
| 
re) 
fe) 
ar 
a 
a 
L— (Tet Te ++ + Te) + Tots) + Tin42) +, &e.)s 
where T(1), T(2), » «+ T(n), &c. now denote merely the terms 
of the feries taken in their order, then becaufe 
T(m42) < = T(m41)> 
I 
T(m+3) < Rie T(m42)> 
Tim+4) < = T(m+3)s 
&c. 
Therefore, 
‘Tm-a) + T (m4 3) F Tm44) Hy Se. < 3 (Tern 
+ T(m42) + Tim+3) +, &c.) 
That 
