[ 75 ] 

 VIII. Methods for the Summation of certain Types of Series. 



By I. J. SCHWATT *. 



I. mO find 

 S=-+ ,,, 1x r + , , ,\, , ^ r*+...= Z 



a a(a + l) a(a + l)(a + 2) W=0 n( a + A;) : 



wherein a is a positive integer, 

 (i.) We may write 



S = (a-1)!2 



n=o (a + n) \ 



_ (a — 1) 1 | r g+n 



_ (a-1)! | r^ 



_ (a— i) ! r;^ _ a 5 1 i!i 



(ii.) A second method is as follows : 



Let u n -\ denote the n\h term of the given series, 



then 



Un _ r 



u n _ 1 a-t-n 9 



and 



or 



Hence 

 and 



2 (a + n) u n =r% u n -i, 



n=\ ?j=1 



00 00 



X (a + n)u n = r%u n+ i. 



n=0 7i = 



»'^r + («-'-)S=l, 



= £[fc- r '- 1 *-+c]. 



* Communicated by the Author. 



