286 NEW SERIES for the 



manifeft, that each term of the feries in our fecond formula 



mull be lefs than — ? of the term before it, but greater than a 



10 



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 aflign limits to the fum of all terms, after 

 any propofed term : for putting it under this form 



f 1 1 + cof a , 1 

 i_ __ J 4 1 — cof a 6 



L — (T(0 + T ( 2) ... + T ( „) + T (n + + T(n+2) -f , &c.)» 



where T(i), T( 2 ), . . . T(„), &c. now denote merely the terms 

 of the feries taken in their order, then becaufe 



T(«+3)<; Y6 T( m +a), 



T(m + 4 )< -g T( OT +3), 



&c. 

 Therefore, 



■T(«-+a) + T(m + 3 ) + T(« + 4 )+, &C. < jg \T(m + i) 

 + T( w + 2) + T(m+3)+, &C.) 



That 



