76 Dr. I. J. Schwatt on Methods for the 



Now 



§e- r r a - 1 dr= — [e- r r a - l + e- r (a—l)r a - 2 



+ e- r (a-lXa-2)r a - 3 +... + e- r (a-l)\ 



a— I r n 



n =on\ 



Therefore (a-l)I-^f- „e* 



r a n=on\ r a 



Multiplying by r a and letting r = Q, we find C = (a — 1) ! 

 We finally obtain 



»-^['-sa- 



II. To find 



S=q-4-^ + 7T-7-F^+^-^^H-...= S 



1.2.3^3.4.5^5.6.7 ^'"~ n to(2n+l){2n + 2)(2n-^Z) 



(i.) Separating the general term into partial fractions 



1 1 r 1 1 1 "I 



(2n + l)(2n + 2)(2n + 3) " 2 L2?i + 1 n + 1 + 271 + 3J' 



therefore 



1 r °° ?> n * ^ w ^ ^ ?i -i 



S= 2 LJo^Tl ""Jo^Tl + Jo^+gJ ' 



= ± fl + r 2 ^~_ j + 2 -^ - 2 -l-r 1 , 



-i[ 1+(1+r >lsS3 + ^^>]' • (A) 



lr 1 + r ^ r*< 2 "+ 8 > 1 1 "I 



But 



„ 3 £ 5. 



9*2 r 3 r a r 2 

 + f -2 + 3-T + 5--"J- 



= | [log (l + ri)- log (l-'f*)] -f* 

 Therefore 



Q 1 ["% , 1 + r, 1 + r* 1 + r 1 /t ,1 



S= ~ 1+ — -log T — 1 + -log(l-r) 



2 L 2r* 1— it* r r ° v 'J 



- r 1-ir-logi r + 21og(l-r)-2 . 



