40 BOOK OF MATHEMATICAL PROBLEMS. 



192. From the equality- 



log (1 -x 3 ) = log (I - x) + log (1 + x + a 2 ), 

 prove that 





ft 



-...}. 



XIII. Recurring Series. 



The series w , u lt u z ...... u n is a recurring series if any fixed 



number (r) of consecutive terms are connected by a relation of the 

 form 



in which w may have any integral value, but p lt P e >"p r - } are 

 constant. It follows that the series a + a 1 x + ajf + . . . + a n x n + . . . 

 is the expansion in ascending powers of x of a function of x of 



, . A + A^x + ... +A r <x r ~ 2 ... 



the form -= a - - (the generating function of 



l+p^ + ptx'* ..,+jp^af ] 



the series); and if the scale of relation (A) and the first r 1 terms 

 of the series be given this function will be completely deter- 

 mined; when, by separating this function into its partial fractions 



7? 7? 



-+z + ...... , and expanding each, we obtain the w th 



term of the series and the sum of n terms. 



If the scale of relation is not given, we shall require 2(r 1) 

 terms of the series to be known to determine all the constants ; 

 thus, if four terms are given we can determine a recurring series 

 with a scale of relation between any three consecutive terms, and 

 whose first four terms are the given quantities. 



