Solution of a Linear Differential Equation. 661 



The given series (1) can be written 



16 /K \ oc „ji 



=^- l) *-i«-i)^ih;> ■ ■ ■ ■ < 9 > 



Then <7S °° **-i 



±UeI1 C Z '^* _ ^ 7 ," + K-l — (11) 



^ »=o 1 — / 



Hence p> B -i - no , 

 S«=J o f^dr+O, (12) 



From (12) follows L 



si= ri^=- iog(i - r); 



C r rdr 



s 3 =J o i^, = - log (i -')-»•; 



S 3 =J j^- r = -log(l-r)-r- g ; 



s ^=J o r^=- 1 og( 1 -'-)-'-2-3 ; 



Q C r t A dr , /1 . r 2 r 3 r 4 



b s=J o i=- r = " l0 S (!-'■) -'- 2 " 3 - 1 ; 



C r i*dr s r 2 r 8 ?' 4 r 5 



S «=J o 1 -^ = -log(l-r)-r- 1 - 3 - i - 5. 



From (10) we obtain : 



= L llog(l-,) + i2 i S i (-l)- 1 A Tiri , (13) 



= ^ ! ^log(l-,)-^j i |(-l>^, (U) 



1 (1-r) 5 , 1 r 137 , 77 47 , 9 1"| 



which is the same result as (7). 



