114 MATHEMATICAL FORMULAE AND ELLIPTIC FUNCTIONS 



6.22 Rummer's transformation. 



AQJ AI, A2, .... is a sequence of positive numbers such that 



Am = A m A m +l , 



a m 

 and 



Limit x 



A TO j 



m-^co 



approaches a definite positive value. Usually this limit can be taken as unity. 

 If not, it is only necessary to divide A m by this limit: 



Limit 



a = A m d m . 



m*<x> 

 Then: 



00 00 f ' 



^a m = (A n a n - a) + 2 ( J ~ A m)0m. 



m=n m=n 



Example i. 



. ^ m Limit ^ 



A m = m, A m = -, A m = i, 



tn + i m 



= 1 



Applying the transformation to the series on the right: 



m -. m 



SI , I 

 ^ = I + r* 



<2 (m -f- i) (m -f 2) 



m=i m=i 



Applying the transformation n times: 



_ _ j 



m 2 ~ 'J m 2 (m + i) (m + 2) . . (m + n) 



Example 2. 



/i T 



A m = -, 



' 



