478 Mr WARBUBTON, ON SELF-REPEATING SERIES. 



2, S, 4, &c. be taken alternately with a positive and with a negative sign. Then since the 

 denominator of the generating fraction is some positive integer power of (l + t), and since this 

 power, when expanded, is a recurrent function of t, the numerator also of the generating frac- 

 tion is a recurrent function of t. 



The numerical values of the coefficient of the numerator, other things remaining the same, 

 will be the same as were obtained in the former case, where the denominator, instead of 

 being a power of (l + t), was a power of (l -f). The numerator in this case will differ from 

 the numerator in the former case, in having the signs of all its alternate terms changed. Thus 

 the fraction which generates (tv + l) 7 , to which corresponds the series, l 7 + 2 7 « + 3~f + &c, being 



1 + 120«+ Il9lf + Hl6f + 119» 4 + 120J 5 + \.f 

 (J-Q 8 ' 



the fraction which generates (-l)*(.i? + l) r , to which corresponds the series 



l 7 - 2 7 * + 3~? - 4 7 * 3 + &C 



will be 



l-120<+1191^-24l6f + 119U 1 - 120f + l.t* 



As in the former case, so in the present, the positive integer powers of the natural numbers, 

 of the odd numbers, and of the figurate numbers of the different orders, are elements from 

 which we may form innumerable other self-repeating series. To this case belongs the particular 



series 



1" _ vt + 3 n f - 4 n < 8 + &c. 



which first drew my attention to the subject now before us ; and I shall proceed to consider 

 this example with some minuteness, for the purpose of shewing that owing to their not being 

 aware of the theorem we are now considering, the distinguished mathematicians who have before 

 directed their attention to this series, have not given to their respective summations all the sim- 

 plicity which the case admits of. 



When the numbers of the natural series are raised to a positive even power, and are taken 

 alternately with opposite signs, the number of the terms in the recurrent numerator being also 

 even, and every two equal recurrent terms having opposite signs, the terms of the numerator, 

 when t is equated to 1, destroy one another, and the value of the generating fraction and of the 

 corresponding infinite series, is zero. 



Thus the generating fraction of l 2 - 2 2 £ + 3 2 f - &c. being ; that of l 4 -2 4 «+3 4 « 2 -&c. 



° (1 + 



1-11*+ \\f -f , „ , . . . -, 



being — ; and so on, all these tractions vanish, when t is equal to 1. 



I proceed therefore to the case where the numbers are raised to an odd power ; and with a 

 view to brevity, I will consider the particular instance of 2ra + 1 = 9. 



Then, if we do not know beforehand that the numerator consists of recurrent terms, the 

 summation, according to the methods in use in the days of Euler, Lagrange, and Laplace, would be 

 made in accordance with the following Diagram, No. I. ; but, if we do know beforehand that 

 the numerator is recurrent, then the summation, founded on the same methods, will be made in 



