278 THIRD REPORT— 1833. 



of triangular numbers to -5-. The series 



1 - 4 + 9 - 16 + 25 - &c. 

 of square numbers to 0. The series of tabular logarithms 

 log 2 — log 3 + log 4 — log 5 + &c., 



would be found to be equal to •0980601 nearly. If we should 

 suppose X negative and greater than 1, the original and the 

 transformed series would become divergent series of the first 

 class. 



The series 



, ,,,(«- \f {a -If {a- \y , 5 



log a = (a - 1) - ^ g ^ + ^ 3 ^ - —^ + &C-- 



is divergent when a is greater than 2, and convertible by Euler's 

 formula into the convergent series 



(«-l) , 1 (« - If ,1 {a-\f \ {a- \f . 



or by the method of Lagrange into the series 



„ (^a _ 1) _ |. (^a _ 1)2 + |- (^_ 1)3 _ &c., 



which may be made to possess any required degree of con- 

 vergency. But it is not necessary to produce further examples 

 of such transformations, which embrace a very great part of the 

 most refined artifices which have been employed in analysis. 



One of the most remarkable of these artifices presents itself 

 in a series to which Legendre has given the name of demicon- 

 vergent*. The factorial function 1^(1 + a:) is expressed by the 

 continuous expression 



(^) '(2*a:)*R, 



where R is a quantity whose Napierian logarithm is expressed 

 by 



A B_ C 



\ .2.x 3 . 4 . x2 ''■ 5 . 6 . ^i" - &c-> 



where A, B, C, &c., are the numbers of Bernoulli. The law of 

 formation of these numbers, as is well known, is extremely 



• Fonetions ElUptiques, torn. ii. chap. ix. p. 425. 



