48 Mr, Horner on ^Jult, 



Article IX. 



On the Use of continued Fractions with unrestricted Numerators 

 in Summation of Series, By W. G. Horner, Esq. 



{Concluded from vol. xi. p. 421.) 



6. Tht word "much" in tlie preceding sentence was written 

 inadvertently : a distinction must be made. It will then appear 

 that the convergency is most promoted where it is most desira- 

 ble ; in the first and second Examples, for instance, more than 

 in the third and 4th. The reason is obvious ; for any series of 

 slow convergency, such as the former two, or indeed any series 

 which varies little in passing from term to term, can ditt'er but 

 little from a recurring series commencing with the same course 

 of terms. 



The mode of estimating the actual degree of approximation 

 has been already noticed. It may not be irrelevant to exem- 

 plify it in this place. The Jirst three terms of the continued 



1 + p 



fraction produce the converging fraction T x a. The nume- 

 rator may be supposed to contain only the first terms of a series 

 1 + P + Q + Sic. indefinitely extended; while the denominator 

 1 + /> is complete. The value of yD being then found from the 

 Equation Q = 0, and substituted in the expression for R 



(Art.2),givesR=~;^>^4^-^a.«, (7) 



for the error of the fourth term in the recurring series equivalent 



. i+P 



to - — . 

 i + p 



The first five terms produce the fraction -^ — —^ x a ; and 



here, as before, the values of j? and q being found from R = 0, 

 and S = 0, produce 



-, n r ~ m s 2 (n r — m s + r s) , ov 



T = — X --^ — T 7— X Oj a^a., (8) 



n + 2 s.n + 3 5 n+3s.n + 4s '^3 ^z 



for the error of the sixth term in the recurring series. 



Example V. — ^Two varieties of the Binomial Theorem become 

 well adapted to arithmetical purposes, by receiving the fractional 

 form, rutting N = the number whose root is to be found, and 



m 



P = the nearest complete power; the first variety is N" = 

 (P + x)" = 



m 



which, after the usual reduction (Art. 4), becomes 



