Summation of certain Types of Series. 77 



(ii.) Another method is the following : 

 If in form (A) of the preceding method we let 



oo A .2?t+3 



r = x 2 and % - — -5 =S 1? 



then jo oo ,,2 



a °i _ ^ # 2 »+ 2 = — - 



d# n=0 1— # 2 ' 



Therefore 

 and 



a C x x 2 dx 1 . 1 -f- # 

 S= ttI l + -log(l — r) H i-loff^ ■ 



=i[^io g i^ +2 iog(i-,)_ 2 ]. 



(iii.) Still another method is the following : 

 Let w n _i designate the nth term of the given series, 

 then u n n(2n-l) 



iin^ (n + l)(2/i+3) ' 



and x 00 



X ( : n + l)(2n + 3)i ()l = rXn(2n-l)u n _ 1 r, 



71=1 11 — 1 



00 



=rt {n + l)(2n+l)u n r 9 



n=0 



and since 1 



(« + l)(2«+3>„1 =i 



-I»=0 * 



therefore 



2(n + l)(2w + 3)M„ = r2(w + l)f2n + l)M ll +^, 

 an( * / d \ T r/ "I 1 



(4, + l)[2(l_^ + 3-,]s=|. 

 Let now • j x 



('£ +1 ) B r'- 



then j ?/ -1 



2(l_,. )r g + (3-,.)» / =i, 



and ., f s-r r* r r s-r 



„—to J2r(l-r) ? t l J2r(l-r) 



Jo r(l-r) 



dr, 



11 — ?' A r* 6?r 



