276 NEW SERIES for the 



where T( m ), T( m + o, T( m + 2 ), &c. denote the terms whofe 

 places in the feries are exprefTed by the numbers m, m-\-i t m-\-2 t 

 &c. Then, becaufe 



T( ff2+2 )< - T( ffl+ i), 

 4 



T (*+3)^ - T( w+2 ), 

 4 



T( CT + 4 ) < - T( w+3 ), 

 4 



&c. 



We have 



T(m+3)4-T( m + 3 ) + T( TO+4 )4', &c. < - fT(„+ x ) 



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



That is, putting S for T (m + 2 ) + T( w+3 ) + T(m +4 )+, & c . or 

 for the fum of. all the terms after T( m +o, 



S<- (t<>-h) + S), and hence ^S < I T(« +I ),& S <- T (w + 1 ). 



4 T" T* 3 



Thus it appears, that the fum of all the terms of the feries fol- 

 lowing any term after the firft, is lefs than the third part of 

 that term. 



11. Again, from what has been faid in Article 9., we have 

 T(« + .)> 3fct3 T ( « + x), and therefore i&M > T(m + l) , 



and fimilarly p^L > 2=!±2> and ^±£) > T(, +3) 



and 



