42 BOOK OF MATHEMATICAL PROBLEMS. 



198. Prove that the series 



3 r + ... +n 



are recurring series, the scales of relation being between 4, 5, ... 

 r 4- 2 terms respectively. 



199. Find the generating functions of the recurring series 



(1) 1 + 2x + 5x 



(2) l + 3x + 4=x* + 



(3) 3 + 6x + Ux 



(4) 3-z+13o; 8 

 and the n* term of each. 



200. If the terms of the series , a,^ a a ... be derived, each 

 from the preceding, by the formula 



prove that 



pq 



XIY. Convergent Fractions. 



201. If '- be the n^ convergent to the continued fraction 



(h a, a,- 



*.+ *.+ "*.+ -. 

 we have the equations 



Pn = &*P*-l + CT n^n_2> 



and the like law for q : and for the fraction 



