slowly convergifig aiid diverging Irifinite Series. 349 



series; and lastly, to exhibit its numerical application in a 

 more commodious form than any in which I have yet seen it. 

 Let a — bx-\-cx^—da^-\-^c, = S 



.*. —bx-^cx'^—dj!^-\-hc, = S—a 



S—a 



b+cx—dx'^ + ea^'^Sic. = 



x-hl 



.'. — 6— Aa?+ A'j;*^— AV + &C. = (S-a) 



.-. -^;a:-^Aa:^ + A'^'-AV + &c. = (a:+l)(S-a) = S^' 



.-. S = -^ +a ; 



that is, 

 ^ bx Ax^ , A^a^ AV ^ 



= a- -^4-_f-/o-A^ + A^x^~AVH-&c.l^ 

 x + \ x + l \ J 



Hence, treating the series within the brackets as we have 

 treated the original, to which it is similar in form, we have 



07 + 1 {x + lf (x + lf'^lx + lY {x + lf^^"^' 

 Similarly, 



bx Ax^ __4^__^l£*_ A!j^__q, 



^-"^ x-^l {x^-lf {x + \y {x + \Y^{x + lY ^°' 



bx Ax^ A'^JT' A«^«+^ 



S = a — 



^+1 (x^-lY (x-\-\f , {J7+1)«+1 



A n+1 ^n+2 



These several expressions for S may be regarded as sq 

 many differential theorems, but the last is that which corre- 

 sponds most nearly to the form usually given : it is, however, 

 more efficient, as it shows that if we stop at the term 



A "^"+^ 

 (.r+l)"+i"' 

 we get one limit to the sum S, and if we stop at the imme- 

 diately following term 



we get another limit, in the contrary sense. These limits, as> 



